11institutetext: Los Alamos National Laboratory, Los Alamos, NM, USA, 48124
11email<EMAIL_ADDRESS>
# Learning Neural Radiance Fields of Forest Structure for Scalable and Fine
Monitoring

Juan Castorena 11

###### Abstract

This work leverages neural radiance fields and remote sensing for forestry
applications. Here, we show neural radiance fields offer a wide range of
possibilities to improve upon existing remote sensing methods in forest
monitoring. We present experiments that demonstrate their potential to: (1)
express fine features of forest 3D structure, (2) fuse available remote
sensing modalities and (3), improve upon 3D structure derived forest metrics.
Altogether, these properties make neural fields an attractive computational
tool with great potential to further advance the scalability and accuracy of
forest monitoring programs.

###### Keywords:

Neural radiance fields Remote Sensing LiDAR ALS TLS Photogrammetry Forestry

##  1 Introduction

With approximately four billion hectares covering around $~{}31\%$ of the
Earth’s land area [7], forests play a vital role in our ecosystem. The
increasing demand for tools that help maintain a balanced and healthy forest
ecosystem is challenging due to the complex nature of various factors,
including resilience against disease and fire, as well as overall forest
health and biodiversity [25]. Active research focuses on the development of
monitoring methods that synergistically collect comprehensive information
about forest ecosystems and utilize it to analyze and generate predictive
models of the characterizing factors. These methods should ideally be capable
of effectively and efficiently cope with the dynamic changes over time and
heterogeneity. The goal is to provide the tools with such properties for
improved planning, management, analysis, and more effective decision-making
processes [1]. Traditional tools for forest monitoring, such as national
forest inventory (NFI) plots, utilize spatial sampling and estimation
techniques to quantify forest cover, growing stock volume, biomass, carbon
balance, and various tree metrics (e.g., diameter at breast height, crown
width, height) [23]. However, these surveying methods consist of manual field
sampling, which tends to introduce bias and poses challenges in terms of
reproducibility. Moreover, this approach is economically costly and time-
consuming, especially when dealing with large spatial extents.

Recent advancements, driven by the integration of remote sensing, geographic
information and modern computational methods, have contributed to the
development of more efficient, cost/time effective, and reproducible ecosystem
characterizations. These advancements have unveiled the potential of highly
refined and detailed models of 3D forest structure. Traditionally, the metrics
collected through standard forest inventory plot surveys have been utilized as
critical inputs in applications in forest health [15], wood harvesting [13],
habitat monitoring [24], and fire modeling [16]. The efficacy of these metrics
relies in their ability to quantitatively represent the full forest’s 3D
structure including its vertical resolution: from the ground, sub-canopy to
the canopy structure. Among the most popular remote sensing techniques,
airborne LiDAR scanning (ALS) has gained widespread interest due to its
ability to rapidly collect precise 3D structural information over large
regional extents [6]. Airborne LiDAR, equipped with accurate position sensors
like RTK (Real-Time Kinematic), enables large-scale mapping from high
altitudes at spatial resolutions ranging from 5-20 points per square meter. It
has proven effective in retrieving important factors in forest inventory plots
[11]. However, it faces challenges in dense areas where the tree canopy
obstructs the LiDAR signal, even with its advanced full-waveform-based
technology. In-situ terrestrial laser scanning (TLS) on the other hand
provides detailed vertical 3D resolution from the ground, sub-canopy and
canopy structure informing about individual trees, shrubs, ground surface, and
near-ground vegetation at even higher spatial resolutions [10]. Recent work by
[20] has demonstrated the efficiency and efficacy of ecosystem monitoring
using single scan in-situ TLS. The technological advances of such models
include new capabilities for rapidly extracting highly detailed quantifiable
predictions of vegetation attributes and treatment effects in near surface,
sub-canopy and canopy composition. However, these models have only been
deployed across spatial domains of a few tens of meters in radius due to the
existing inherited limitations of TLS spatial coverage [20]. On the other side
of the spectrum, image based photogrammetry for 3D structure extraction offers
the potential of being both scalable and the most cost efficient. Existing
computational methods for the extraction of 3D structure in forest ecosystems,
however, have not been as efficient. Aerial photogrammetry methods result in
3D structure that contains very limited structural information along the
vertical dimension and have encountered output spatial resolutions that can be
at most only on par with those from ALS [25].

Our contribution seeks to fuse the experimental findings across remote sensing
domains in forestry; from broad-scale to in-situ sensing sources. The goal is
the ability to achieve the performance quality of in-situ sources (e.g., TLS)
in the extraction of 3D forest structure at the scalability of broad sources
(e.g., ALS, aerial-imagery). We propose the use of neural radiance field
(NERF) representations [17] which account for the origin and direction of
radiance to determine highly detailed 3D structure via view-consistency. We
observe that such representations enable both the fine description of forest
3D structure and also the fusion of multi-view multi-modal sensing sources.
Demonstrated experiments on real multi-view RGB imagery, ALS and TLS validate
the fine resolution capabilities of such representations as applied to
forests. In addition, the performance found in our experiments of 3D structure
derived forest factor metrics demonstrate the potential of neural fields to
improve upon the existing forest monitoring programs. To the best of our
knowledge, the demonstrations conducted in this research, namely, the
application of neural fields for 3D sensing in forestry, is novel and has not
been shown previously. In the following, Sec. 2 provides a brief overview of
neural fields. Sec. 3 includes experiments illustrating the feasibility of
neural fields to represent fine 3D structure of forestry while Section 4
demonstrates the effectiveness of fusing NERF with LiDAR data by enforcing
LiDAR point cloud priors. Finally, Section 5 presents results that show the
efficacy of NERF extracted 3D structure for deriving forest factor metrics,
which are of prime significance to forest managers for monitoring.

##  2 Background

###  2.1 Neural Radiance Fields

The idea of neural radiance fields (NERF) is based on classical ray tracing of
volume densities [12]. Under this framework, each pixel comprising an image is
represented by a ray of light casted onto the scene. The ray of light is
described by $\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}$ with origin
$\mathbf{o}\in\mathbb{R}^{3}$, unit $\ell_{2}$-norm length direction
$\mathbf{d}\in\mathbb{R}^{3}$ (i.e., $\|\mathbf{d}\|_{2}=1$) and independent
variable $t\in\mathbb{R}$ representing a relative distance. The parameters of
each ray can be computed through the camera intrinsic matrix $\mathbf{K}$ with
inverse $\mathbf{K}^{-1}$, the 6D pose transformation matrix
$\mathbf{T}_{m\rightarrow 0}$ of image $m$ as in Eq. (2.1)

$\displaystyle(\mathbf{o},\mathbf{d})=\left(T^{(4)}_{m\rightarrow
0},\frac{\mathbf{d}^{\prime}}{\|\mathbf{d}^{\prime}\|_{\ell_{2}}}\right)\quad\text{with
}$ $\displaystyle\mathbf{d}^{\prime}=\mathbf{T}^{-1}_{m\rightarrow
0}\mathbf{K}^{-1}\begin{bmatrix}u^{\prime}\\\ v^{\prime}\\\
1\end{bmatrix}-T^{(4)}_{m\rightarrow 0}$ (1)

where $u^{\prime},v^{\prime}$ are vertical and horizontal the pixel locations
within the image and the subscript (i) denotes the $i$-th column of a matrix.
Casting rays $\mathbf{r}\in\mathcal{R}$ into the scene from all pixels across
all multi-view images provides information of intersecting rays that can be
exploited to infer 3D scene structure. Such information consists on sampling
along a ray at distance samples $\\{t_{i}\\}_{i=1}^{M}$ and determine at each
sample if the color $\mathbf{c}_{i}\in[0,..,255]^{3}$ of the ray coincides
with those from overlapping rays. If it does not coincide then it is likely
that the medium found at that specific distance sample is transparent whereas
the opposite means an opaque medium is present. With such information,
compositing color can be expressed as a function of ray $\mathbf{r}$ as in Eq.
(2) by:

$\hat{C}(\mathbf{r})=\sum_{i=1}^{N}\left[\underbrace{\left(\prod_{j=1}^{i-1}\exp(-\sigma_{j}\delta_{j})\right)}_{\text{transparency
so
far}}\underbrace{(1-\exp(-\sigma_{i}\delta_{i}))}_{\text{opacity}}\mathbf{c}_{i}\right]$
(2)

where $\sigma_{i}\in\mathbb{R}$ and $\delta_{i}=t_{i+1}-t_{i}$ are the volume
densities and differential time steps at sample indexed by $i$, respectively.
In Eq. (2) the first term in the summation represents the transparent samples
so far while the second term is an opaque medium of color $\mathbf{c}_{i}$
present at sample $i$. Reconstructing a scene in 3D can then be posed as the
problem of finding the sample locations $t_{i}$ where each ray intersects an
opaque medium (i.e., where each ray stops ) for all rays casted into the
scene. Those intersections are likely to occur at the sample locations where
the volume densities are maximized; in other words, where
$t_{i}=\mathop{\mathrm{arg\,max}}_{i}\\{\mathbf{\sigma}\\}$. Accumulating, all
rays casted into the scene and estimating the locations $t_{i}$’s where volume
density is maximized overall rays, renders the 3D geometry of the scene. The
number of rays required per scene is an open question; the interested reader
can go to [3] where a similar problem but for LiDAR sensing determines the
number of pulses required for 3D reconstruction depending on a quantifiable
measure of scene complexity.

The problem in Eq. (2) is solved by learning the volume densities that best
explains image pixel color in a 3D consistent way. Learning can be done
through a multilayer perceptron (MLP) by rewriting Eq. (2) as in Eq. (3) as:

$\hat{C}(\mathbf{r})=\sum_{i=1}^{N}\mathbf{w}_{i}\mathbf{c}_{i}$ (3)

where the weights $\mathbf{w}\in\mathbb{R}^{N}$ encode transparency or opacity
of the $N$ samples along a ray and $\mathbf{c}_{i}$ is its associated pixel
color. Learning weights is performed in an unsupervised fashion through the
optimization of a loss function using a training set of $M$ pairs of multi-
view RGB images and its corresponding 6D poses
$\\{(\mathbf{y}_{m},\mathbf{T}_{m})\\}_{m=1}^{M}$, respectively. This loss
function $f:\mathbb{R}^{L}\rightarrow\mathbb{R}$ is the average
$\ell_{2}$-norm error between ground truth color and estimation by compositing
described as in Eq. (4):

$\pazocal{L}_{C}(\mathbf{\Theta})=\sum_{\mathbf{r}\in\mathcal{R}}\left[\|C(\mathbf{r})-\hat{C}(\mathbf{r},\mathbf{\Theta})\|_{\ell_{2}}^{2}\right]$
(4)

Optimization by back-propagation yields the weights that gradually improves
upon the estimation of the volume densities. Other important parameters of
NERF are distance $\hat{z}(\mathbf{r})$ which can be defined using the same
weights from Eq.(2) but here expressed in terms of distance as:

$\hat{z}(\mathbf{r})=\sum_{i=1}^{N}\omega_{i}t_{i},\quad\qquad\hat{s}(\mathbf{r})^{2}=\sum_{i=1}^{N}\omega_{i}(t_{i}-\hat{z}(\mathbf{r}))^{2}$
(5)

and $\hat{s}(\mathbf{r})$ defined as the standard deviation of distance. One
key issue affecting 3D reconstruction resolution is on the way samples
$\\{t_{i}\\}_{i=1}^{N}$ for each ray $\mathbf{r}\in\mathcal{R}$ are drawn. A
small number of samples $N$ results in low resolution and erroneous ray
intersection estimations while sampling vastly results in much higher
computational complexities. To balance this trade-off, the work in [17] uses
two networks one at low-resolution to coarsely sample the 3D scene and another
fine-resolution one used subsequently to more finely sample only at locations
likely containing the scene.

##  3 Are neural fields capable of extracting 3D structure in forestry?

The high capacity of deep learning (DL) models to express data distributions
with high fidelity and diversity offers a promising avenue to model
heterogenous 3D forest structures in fine detail. The specific configuration
of the selected DL model aims to provide a representation that naturally
allows the combination of data from multiple sensing modalities and view-
points. Neural fields [17] under the DL rubric have proven to be a highly
effective computational approach for addressing such problems. However, their
application has been only demonstrated for indoor and urban environments.

###  3.1  Terrestrial Imagery.

Expanding on the findings of neural fields in man-made environments, we
conducted additional experiments to demonstrate its effectiveness in
representing fine 3D structure details in forest ecosystems. Figs. 1 and 2
shows the extracted 3D structure of a Ponderosa pine tree in New Mexico,
captured using standard 12-megapixel camera phone images collected along an
elliptical trajectory around the tree. Fig. 1(a) shows a few of the input
example terrestrial multi-view RGB images collected. Figs. 1(b) and1(c)
presents the image snapshot trajectory represented as red rectangles, along
with two 3D structure views derived from a traditional structure from motion
(SFM) method [22] applied to the multi-view input images. Note that the level
of spatial variability detail provided by this SFM method is significantly low
considering the resolution provided by the set of input images.

(a) Terrestrial RGB multi-view imagery of Ponderosa Pine Tree.

(b) SFM reconstruction view-1

(c) SFM reconstruction view-2

Figure 1: Even though SFM reconstruction is capable of extracting the 3D
structure of tree, its recontruction suffers from sparsity. Such sparsity
limits the spatial variability of structure that can be captured thorugh such
models.

Can the representational power of modern AI models do better than classical 3D
structure extraction methods in Forestry? We extract 3D structure by neural
fields using the same input images and obtain the result shown in Fig.2. Note
that much finer spatial variability details can be resolved across the 3D
structure including the ground, trunk, branches, leaves. Even fine woody
debris as shown in Figs2(c)-.2(d) and, bark can be resolved as shown in
Figs.2(e)-2(f) in contrast to the result of traditional SFM in Fig.1. Note
that even points coming from images degraded by sun-glare as shown in Fig.1(a)
landed in the tree within reasonable distances as shown in Fig.2(a), this is
significant specially considering the severity of the glare effects present in
the 2D RGB images. In general, terrestrial multi-view imagery based NERF can
be used to extract fine 3D spatial resolution along the vertical dimension of
a tree stand with a level of detail similar to TLS and with the additional
advantage of providing color for every 3D point estimate.

(a) Side views illustrating high 3D spatial detail along the vertical tree
stem

(b) Tree 3D structure view-5

(c) Fine 3D resolution of forest floor structure

(d) Forest floor 3D structure

(e) Tree trunk view-1

(f) Tree trunk view-2

Figure 2: Neural field models are capable of extracting fine 3D structure from
terrestrial multi-view images in forestry. Reconstructions demonstrate their
potential to represent fine scale variability in heterogeneous forest
ecosystems.

##  4 Neural Radiance Fields: A framework for remote sensing fusion in
forestry

Neural fields, have also demonstrated their ability to provide representations
suitable for combining data from multiple sensing modalities in as long as
these are co-registered or aligned. The neural fields framework, which
extracts 3D structure from multi-view images, enables direct fusion of
information with 3D point cloud sources through point cloud prior constraints
[21]. Here, we consider the case of fusing multi-view images from an RGB
camera and point clouds from LiDAR. The difficulty in fusing camera and LiDAR
information is that camera measures color radiance while LiDAR measures
distance [5]. Fortunately, the framework of neural radiance fields can be used
to extract 3D structure from images thus enabling direct fusion of information
from LiDAR. This can be done though a learning function that extracts a 3D
structure promoting consistency between the multi-view images as leveraged by
standard NERF [17] subject to LiDAR point cloud priors [21] as:

$\pazocal{L}(\mathbf{\Theta})=\underbrace{\sum_{\mathbf{r}\in\mathcal{R}}\left[\|C(\mathbf{r})-\hat{C}(\mathbf{r},\mathbf{\Theta})\|_{\ell_{2}}^{2}\right]}_{\pazocal{L}_{C}(\mathbf{\Theta})}+\lambda\underbrace{\sum_{\mathbf{r}\in\mathcal{R}}\left[\|z(\mathbf{r})-\hat{z}(\mathbf{r},\mathbf{\Theta})\|_{\ell_{2}}^{2}\right]}_{\pazocal{L}_{D}(\mathbf{\Theta})}$
(6)

where the first term $\pazocal{L}_{C}(\mathbf{\Theta})$ is the standard NERF
learning function promoting a 3D structure with consistency between image
views while the second term $\pazocal{L}_{D}(\mathbf{\Theta})$ enforces the
LiDAR point cloud priors with $\hat{z}(\mathbf{r},\mathbf{\Theta})$ given as
in Eq.(5). The benefit of imposing point cloud priors into neural fields is
two-fold: (1) it enables expressing relative distances obtained from standard
3D reconstruction of multi-view 2D images in terms of real metrics (e.g.,
meters), and (2) neural fields tend to face challenges in accurately
estimating 3D structures at high distances (typically in the order of several
tens of meters), where the LiDAR point cloud priors can serve as a supervisory
signal to guide accurate estimation, especially at greater distances. This can
be beneficial, as distances in aerial imagery are generally distributed around
large distances, which may pose challenges for 3D structure extraction
methods.

###  4.1 Filing in the missing below-canopy structure in ALS data with TLS

In-situ terrestrial laser scanning (TLS) has been demonstrated as a powerful
tool for rapid assessment of forest structure in ecosystem monitoring and
characterization. It is capable of very fine resolution including the vertical
direction: surface, sub-canopy and canopy structure. However, its utility and
application is restricted by limited spatial coverage. Aerial laser scanning
(ALS) on the other hand, has the ability to rapidly survey broad scale areas
at the landscape level, but is limited as it sparsely samples the scene
providing only coarse spatial variability details and it also cannot penetrate
the tree canopy. Fig. 3(a) shows a point cloud example collected using a full-
waveform ALS system which collects $\approx$ 10 points per meter square. In
Fig. 3(a) note that the sub-canopy structure is not spatially resolved. In
contrast, TLS is finely resolved below the canopy as observed in Fig.3(b).

(a) ALS side-view

(b) TLS side-view

Figure 3: Forest structure from TLS and ALS: ALS provides sparse spatial
information and is not capable of resolving sub-canopy detail. TLS on the
other hand, provides fine spatial variability and resolution along full 3D
vertical stands.

Fortunately, the drawbacks of TLS and ALS scans can be resolved by co-
registration which transforms the data to enable direct fusion. Here, we use
the automatic and targetless based approach of [4]. This was demonstrated to
outperform standard methods [2], [19], [8] in natural ecosystems and to be
robust to resolution scales, view-points, scan area overlap, vegetation
heterogeneity, topography and to ecosystem changes induced by pre/post low-
intensity fire effects. It is also fully automatic, capable of self-correcting
in cases of noisy GPS measurements and does not require any manually placed
targets [9] while performing at the same levels of accuracy. All TLS scans
where co-registered into the coordinate system of ALS. Once scans have been
co-registered they can be projected into a common coordinate system.
Illustrative example results for two forest plots where included in Fig.4
where the two sources: ALS and TLS have been color coded differently, with the
sparser point cloud being that of the ALS. Throughout all cases the co-
registration produced finely aligned point clouds.

(a) Co-registration Example 1

(b) Co-registration Example 2

Figure 4: TLS to ALS co-registration: Forest features are well aligned
qualitatively between both ALS and TLS sensing.

In general, the error produced by this co-registration method is $<$6 cm for
the translation and $<$0.1${}^{\text{o}}$ for the rotation parameters. The
translation error in mainly due to the resolution of ALS at 10 points/meter
square.

###  4.2 Aerial Imagery

Experiments performed on broader forest areas were also conducted. Aerial RGB
imagery was collected with a DJI Mavic2 Pro drone at 30Hz and a 3840 $\times$
2160 pixel resolution. Figs. 5(a)-5(f) show examples of multi-view aerial
image inputs used by the SFM and neural fields models. The forest 3D structure
resulting from running conventional SFM [22] on these images is in
Figs.5(i)-5(k) illustrating different perspective views. Again, the sequence
of rectangles in red illustrate the drone flight path and the snapshot image
locations. Note that SFM was capable of resolving 3D structure for the entire
scene.

Applying NERF directly into the RGB imagery dataset, did not result in
comparable performance as in the case of the Ponderosa pine tree shown in
Section 3.1. Without point cloud constraints, the 3D structure extracted by
the neural fields in Fig. 5(h) shows the presence of artifacts at large
distances. The main reason for these artifacts is that NERF had difficulties
in recovering 3D structures from images with objects distributed at far
distances (e.g., ground surface in aerial scanning). Imposing LiDAR point
cloud priors we hypothesize can help to alleviate this issue. Here, we follow
the methodology of [21] and conduct experiments for fusing camera and LiDAR
information through the learning function in Eq.(6). The LiDAR point cloud
uses both co-registered TLS and ALS data which provides information to
constrain both distances in the mid-story below the canopy and those between
the ground surface and the tree canopy. The co-registration approach used to
align ALS and TLS point clouds is the one described in Section 4.1. Note that
TLS information is not available throughout the entire tested forest area;
rather, only one TLS scan was collected. We found the information provided by
just one single scan was enough to constraint the relative distances in sub-
canopy areas throughout the entire scene. Imposing additional constraints
through consistency with the input point cloud shown in Fig.5(g), results in
the extracted 3D structure shown in Figs. 5(l)-5(n). In this case, the point
cloud prior imposes constraints that resolve the associated difficulties at
large distances. Note that this reconstruction is significantly less sparser
than those shown in Figs.5(i)-5(j) obtained from conventional SFM. NERF+LIDAR
results in improved resolution which in turn enables the detection of a much
finer spatial variability, specially important for current existing demands in
forest monitoring at broad scale. This illustrates the capacity of neural
fields models not only to represent highly detailed 3D forest structure from
aerial multi-view data but also the possibility of combining multi-source
remotely sensed data (i.e., imagery and LiDAR).

(a) Image view-1

(b) Image view-2

(c) Image view-3

(d) Image view-4

(e) Image view-5

(f) Image view-6

(g) Point Cloud

(h) NERF artifacts

(i) COLMAP view-1

(j) COLMAP view-2

(k) COLMAP view-3

(l) NERF+LIDAR view-1

(m) NERF+LIDAR view-2

(n) NERF+LIDAR view-3

Figure 5: AI-based extraction of 3D structures from aerial multi-view 2D
images + 3D point cloud data inputs. Imposing point cloud priors into 3D
structure extraction improves distance ambiguities in structure and resolves
artifact issues likely at far ranges.

##  5 Prediction of forest factor metrics

Demonstration of the described capabilities of neural fields on forest
monitoring programs consists here in performance evaluations of 3D forest
structure derived metrics. These can include for example number of trees,
species composition, tree height, diameter at breast height (DBH), age on a
given geo-referenced area. However, since our focus is to demonstrate the
usefulness of neural radiance fields for representing 3D forest structure, we
only illustrate its potential in prediction of the number of trees and DBH
along geo-referenced areas. The data used includes overlapping
TLS+ALS+GPS+aerial imagery multi-view multi-modal data collected over forest
plot units. Each of these plots represents a location area of a varying size:
some of size 20 x 50 m and others at 15 m radius. The sites in which data was
collected is in northern New Mexico, USA (the NM dataset). The vegetation
heterogeneity and topography variability of the landscape is significantly
diverse. The NM site contains high elevation ponderosa pine and mixed-conifer
forest: white fir, limber pine, aspen, Douglas fir and Gambel oak and
topography is at high elevation and of high-variation (between 5,000-10,200
ft). The TLS data was collected using a LiDAR sensor mounted on a static
tripod placed at the center of each plot. The ALS data was collected by a
Galaxy T2000 LiDAR sensor mounted on a fixed-wing aircraft. The number of
LiDAR point returns per volume depend on the sensor and scanning protocol
settings (e.g., TLS or ALS, range distribution, number of scans) and these
vary across plots depending on the heterogeneity of the site.

(a) In-situ plot-scale TLS has demonstrated to be an effective tool in
estimating plot-level vegetation characteristics

(b) Broad-landscape scale ALS derived prediction, does not have vertical
dimension resolution resulting in underestimate predictions

Figure 6:  LiDAR derived vegetation attribute estimation for single TLS and
ALS.

Ground truth number of trees per plot was obtained through standard forest
plot field surveying techniques involving actual physical measurements of
live/dead vegetation composition. Data from a total of 250 plots where
collected in the NM dataset. In every forest plot overlapping ALS, GPS, TLS
and multi-view aerial imagery data was collected along with the corresponding
field measured ground truth. Prediction of the number of trees $y_{1}$ per
plot given point cloud $\mathbf{X}$, was performed following the approach of
the GRNet [27] [26]. In general, the methodology consists in computing 2D
bounding boxes each corresponding to a tree detection from a birds eye view
(BEV). A refinement segmentation approach then follows which projects each 2D
bounding box into 3D space. The resulting points inside each 3D bounding box
are then segmented by foliage, upper stem and lower stem and empty space and
this information is used to improve estimates over the number of trees. This
methodology is used independently on several case scenarios comparing the
performance of a combination of remote sensing approaches: (1) neural fields
(NF) from aerial RGB Images, (2) ALS as in Fig.6(b), (3) TLS as in Fig.6(a),
(4) ALS+TLS, (5) NF-RGB images + ALS, (6) NF-RGB images + TLS, (7) NF-RGB
Images + TLS + ALS. Note that the TLS, ALS and TLS+ALS prediction results does
not make any use of neural fields. Rather, their performance was included only
for comparison purposes. Table 1 summarizes the root mean squared error (RMSE)
results for each of the tested cases.

Table 1: RMSE Prediction performance of number of trees per plot in NM dataset. Method | NF-RGB | ALS | TLS | ALS+TLS | NF-RGB+ALS | NF-RGB+TLS | NF-RGB+ALS+TLS
---|---|---|---|---|---|---|---
RMSE | 10.61 | 8.44 | 1.77 | 1.67 | 1.41 | 1.39 | 1.32

The results in Table 1 corroborate some of the trade-offs between the sensing
modalities and in addition some of the advantages gained through the use of
neural fields in forestry. First, the superiority of TLS over ALS data on the
number of trees metric is mainly due to the presence of information in sub-
canopy which is characteristic of in-situ TLS. This in alignment with current
demonstrations in the literature which have motivated the widespread usage of
in-situ TLS in forestry applications even though it is not as spatially
scalable as ALS is [20]. We would have seen the opposite relationships between
TLS and ALS, however, in cases when the plot size is significantly higher than
the range of a single in-situ TLS scan. A problem which can be resolved by
adding multiple view co-registered TLS scans per plot. This limitation is
caused as the sensor remains static at collection time which makes it more
susceptible to occlusions, specially in dense forest areas where trees can
significantly reduce the view of TLS at higher ranges. TLS+ALS overcomes, on
the other hand, the limitations of the individual LiDAR platforms by filling
in the missing information characteristic of each platform. Structure from
neural fields using only multi-view RGB images performed slightly worst than
both ALS and TLS. This may be due to the limited number of multi-view images
collected per plot, the performance for deriving structure from NERF or to the
joint performance of NERF in conjunction with the GRNet. Fortunately, fusing
neural fields from multi-view imagery with LiDAR shows a significant
improvement overall fused cases (i.e., NF+ALS, NF+TLS and NF+ALS+TLS). We see
that the prior supervisory signal imposed by the LiDAR point cloud helps on
guiding the resulting 3D structure from NERF to alleviate the artifacts
arising at far distances when using multi-view imagery only. We would like to
finalize this discussion by highlighting the performance of the NF-RGB+ALS
method which is marginally similar to the best performing method (i.e., NF-
RGB+ALS+TLS). The benefit of using NF-RGB+ALS is that being both airborne
makes the data collection of these two modalities time and cost efficient, in
contrast, to in-situ remote sensing methods such as TLS. This has significant
implications towards achieving both scalable and highly performing forest
monitoring programs. In general, one has to resort to a balance between
scalability and performance performance depending on needs. Our work instead,
offers a method which can potentially achieve similar performance as in-situ
methods with the benefits of scalability over the landscape scales through
computational methods.

Additional experiments were conducted to explore the ability of neural fields
from terrestrial based multi-view imagery to achieve a performance near that
of TLS in metrics that depend on sub-canopy information. In this case, we
evaluated performance on the DBH metric for a total of 200 trees. Ground truth
DBH was manually measured in the field for each tree’s stem diameter at a
height of 1.3m. A total of 5 co-registered TLS scans where used per tree, each
collected from a different location and viewing each tree from a different
perspective to reduce the effects of occlusion and to remove the degrading
effects of lower point LiDAR return densities at farther ranges. Multi-view
TLS co-registration was obtained using the method of [4]. Terrestrial multi-
view RGB imagery data for NERF was collected around an oblique trajectory
around each tree as exemplified in Fig.1 with $10-15$ snapshot images per
tree. Algorithmic performance for estimating DBH was compared against TLS,
ALS, TLS+ALS and NF-RGB. The estimation approach of [26] relying on stem
geometric circular shape fitting at a height of 1.3m over the ground was used
following their implementation. Performance is measured as the average error
as a percentage of the actual field measured DBH ground truth, following the
work of [26]. Comparison results are reported in Table 2.

Table 2: Comparison of sensing modalities on average error DBH estimation. Method | NF-RGB | ALS | TLS | ALS+TLS
---|---|---|---|---
Avg. error $\%$ | 1.7 $\%$ | 32.7$\%$ | 1.3$\%$ | 3.3$\%$

In table 2 ALS performs the worst DBH estimation due to its inherited limited
sub-canopy resolution. Multi-view TLS on the other hand, performs the best at
1.3$\%$ error consistent with TLS superiority findings in [26] for metrics
relying on sub-canopy information. However, our neural fields approach from
terrestrial imagery performs marginally on par with multi-view TLS, with the
additional advantage that RGB camera sensors are simpler to access
commercially and significantly cheaper than LiDAR.

In terms of computational specifications, neural radiance fields were trained
using a set of overlapping 10-50 multi-view images per scene. The fast
implementation of [18] was used with training on the terrestrial and aerial
multi-view imagery taking from 30-60 secs per 3D structure extraction (e.g.,
per plot in the aerial imagery case, per tree in the terrestrial imagery
case). Adding the LiDAR constraints was done following the implementation from
[21]. The neural radiance architecture is a multilayer perceptron (MLP) with
two hidden layers and a ReLU layer per hidden layer and a linear output layer
as in [18]. Training was performed using the ADAM optimizer [14] with
parameters $\beta_{1}$ = 0.9, $\beta_{2}$ = 0.99, $\epsilon=10^{-15}$ using
NVIDIA Tesla V100.

The main limitation of neural fields from aerial multi-view imagery is the
presence of occlusion of sub-canopy structure, specially in densely forested
areas. In our case, fusion with TLS data can resolve this problem as
terrestrial data provides highly detailed sub-canopy information.
Additionally, when TLS is unavailable, terrestrial imagery can be used
instead. Our 3D structure experiments from terrestrial multi-view information
in Sec.3.1 and the DBH estimation performance results demonstrate that highly
detailed structure along the entire vertical stand direction can be extracted
by neural fields when image information is available. In the absence of multi-
view image data, however, neural fields are not capable of generating
synthetic information behind occluded areas and performance on metrics
affected by occlusion are expected to yield large errors. This problem can be
alleviated through multi-view images capturing the desired areas of interest
in the ecosystem.

##  6 Conclusion

In this work, we proposed neural radiance fields as representations that can
finely express the 3D structure of forests both in the in-situ and at the
broad landscape scale. In addition, the properties of neural radiance fields;
in particular, the fact that they account for both the origin and direction of
radiance to define 3D structure enables the fusion of data coming from
multiple locations and modalities; more specifically those from multi-view
LiDAR’s and cameras. Finally, we evaluated the performance of 3D structure
derived metrics typically used in forest monitoring programs and demonstrated
the potential of neural fields to improve performance of scalable methods at
near the level of in-situ methods. This not only represents a benefit on
sampling time efficiency but also has powerful implications on reducing
monitoring costs.

## Acknowledgements

Research presented in this article was supported by the Laboratory Directed
Research and Development program of Los Alamos National Laboratory under
project number GRR0CSRN.

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# Domains without parabolic minimal submanifolds and
weakly hyperbolic domains

Franc Forstnerič  Franc Forstnerič, Faculty of Mathematics and Physics,
University of Ljubljana, and Institute of Mathematics, Physics, and Mechanics,
Jadranska 19, 1000 Ljubljana, Slovenia<EMAIL_ADDRESS>
(Date: 1 June 2023)

###### Abstract.

We show that if $\Omega$ is an $m$-convex domain in $\mathbb{R}^{n}$ for some
$2\leq m<n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive
radius and is not $m$-flat near infinity, then $\Omega$ does not contain any
immersed parabolic minimal submanifolds of dimension $\geq m$. In particular,
if $M$ is a properly embedded nonflat minimal hypersurface in $\mathbb{R}^{n}$
with a tubular neighbourhood of positive radius then every immersed parabolic
hypersurface in $\mathbb{R}^{n}$ intersects $M$. In dimension $n=3$ this holds
if $M$ has bounded Gaussian curvature function. We also introduce the class of
weakly hyperbolic domains $\Omega$ in $\mathbb{R}^{n}$, characterised by the
property that every conformal harmonic map $\mathbb{C}\to\Omega$ is constant,
and we elucidate their relationship with hyperbolic domains, and domains
without parabolic minimal surfaces.

###### Key words and phrases:

minimal surface, $m$-plurisubharmonic function, hyperbolic domain

###### 2010 Mathematics Subject Classification:

Primary 53A10. Secondary 32Q45

Supported by the European Union (ERC Advanced grant HPDR, 101053085 to F.
Forstnerič) and by grants P1-0291, J1-3005, and N1-0237 from ARRS, Republic of
Slovenia.

##  1\. Introduction

This paper is motivated by two closely related lines of developments in the
theory of minimal surfaces. The first one is the circle of results known as
halfspace theorems. The second one is the recently introduced hyperbolicity
theory for minimal surfaces.

Concerning the first topic, Xavier [42] proved in 1984 that the convex hull of
a complete nonflat minimal surface in $\mathbb{R}^{3}$ with bounded Gaussian
curvature equals $\mathbb{R}^{3}$. The Strong Halfspace Theorem of Hoffman and
Meeks [23, Theorem 2] says that any two proper minimal surfaces in
$\mathbb{R}^{3}$ intersect, unless they are parallel planes. By Pacelli Bessa,
Jorge, and Oliveira-Filho [8] and Rosenberg [40], the same conclusion holds
for a pair of complete immersed minimal surfaces in $\mathbb{R}^{3}$ with
bounded curvature; such surfaces need not be proper in $\mathbb{R}^{3}$ (see
Andrade [6]) unless they are embedded (see Meeks and Rosenberg [38, Theorem
2.1]). Closer to the topic of this paper, Pacelli Bessa, Jorge, and Pessoa
proved in [9] that an immersed parabolic minimal surface in $\mathbb{R}^{3}$
intersects every complete immersed nonflat minimal surface with bounded
curvature. Related developments concern the maximum principle at infinity; see
the surveys by Meeks and Pérez [32, 34] and the recent paper by Gama et al.
[17].

In this paper we find geometric conditions on the boundary of a domain
$\Omega$ in $\mathbb{R}^{n}$ for $n\geq 3$ implying that $\Omega$ does not
contain any immersed (not necessarily proper or complete) parabolic minimal
submanifolds of a given dimension; see Theorem 2.1. In particular, we obtain a
halfspace theorem for a pair of minimal hypersurfaces in $\mathbb{R}^{n}$, one
of which is parabolic and the other one is properly embedded and has a tubular
neighbourhood of positive radius; see Corollary 2.2. When $n=3$, this
coincides with a special case of the aforementioned result of Pacelli Bessa et
al. [9, Theorem 1.1]; see Corollary 2.3.

Parabolicity of an open Riemannian manifold is important from the point of
view of potential theory. Determining whether such a manifold is parabolic or
hyperbolic is a classical question, the so-called type problem. The survey by
Grigor’yan [19] is a standard reference. The extrinsic case, considering the
type problem for a submanifold of a Riemannian manifold, is natural and
interesting as well, in particular when the ambient manifold is a Euclidean
space. The case of minimal surfaces in $\mathbb{R}^{n}$ has been widely
studied. For minimal submanifolds of dimension higher than $2$ there is not
much literature, the reason probably being that every complete minimal
submanifold of dimension at least $3$ in a Euclidean space is hyperbolic; see
Markvorsen and Palmer [31, Theorem 2.1].

The absence of parabolic minimal submanifolds in a given domain indicates that
the domain is small or tight for minimal submanifolds. Another measure of
smallness is hyperbolicity (as measured by minimal surfaces), a notion
introduced in the recent paper by Forstnerič and Kalaj [16], and developed
further by Drinovec Drnovšek and Forstnerič [12]. A domain
$\Omega\subset\mathbb{R}^{n}$ is hyperbolic if for every point $p\in\Omega$
there are a neighbourhood $U\subset\Omega$ of $p$ and a constant
$c_{p}<\infty$ such that every conformal harmonic disc
$f:\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}\to\Omega$ with $f(0)\in U$ satisfies
$\|df_{0}\|\leq c_{p}$. This is a close analogue of Kobayashi hyperbolicity in
complex analysis; see Kobayashi [27, 28]. In Section 4 we introduce the class
of weakly hyperbolic domains — domains $\Omega\subset\mathbb{R}^{n}$ $(n\geq
3)$ without nonconstant conformal harmonic maps $\mathbb{C}\to\Omega$. Weak
hyperbolicity is an analogue of Brody hyperbolicity, which excludes
nonconstant holomorphic maps from $\mathbb{C}$ to a given complex manifold. A
hyperbolic domain is also weakly hyperbolic; the converse holds on convex
domains (see Proposition 4.4) but fails in general. In fact, we show that the
class of weakly hyperbolic domains properly contains the other two mentioned
classes, and there is a nonhyperbolic domain in $\mathbb{R}^{3}$ without
parabolic minimal surfaces (see Proposition 4.2). On the other hand, it
remains an open problem whether there exists a hyperbolic domain containing a
parabolic minimal surface.

##  2\. Excluding parabolic minimal submanifolds

Let $m\geq 2$ be an integer. A connected $m$-dimensional Riemannian manifold
$(R,g)$ is said to be parabolic if every negative subharmonic function on $R$
is constant. A connected manifold immersed into a Euclidean space
$\mathbb{R}^{n}$ is said to be parabolic if it is parabolic in the metric
induced from the Euclidean metric on $\mathbb{R}^{n}$ by the immersion. On a
surface $R$, parabolicity only depends on the conformal class of the metric.
Every compact conformal surface punctured at finitely many points is
parabolic; an example is $\mathbb{C}=\mathbb{CP}^{1}\setminus\\{\infty\\}$.
See Grigor′yan [19] for more information. The following result is proved in
Section 3.

######  Theorem 2.1.

Let $2\leq m<n$ be integers. Assume that $\Omega$ is a domain in
$\mathbb{R}^{n}$ with $\mathscr{C}^{2}$ boundary $M=b\Omega$ whose principal
curvatures $\nu_{1}\leq\nu_{2}\leq\cdots\leq\nu_{n-1}$ satisfy
$\nu_{1}+\nu_{2}+\cdots+\nu_{m}\geq 0$ at every point, the set of $m$-flat
points $\\{p\in M:\nu_{j}(p)=0\ \text{for}\ j=1,\ldots,m\\}$ has bounded
interior in $M$, and there is an $\epsilon>0$ such that every point in the
$\epsilon$-neighbourhood of $M$ has a unique nearest point in $M$. Then,
$\Omega$ does not contain any parabolic immersed minimal submanifolds of
dimension $\geq m$. In particular, if these conditions hold for $m=2$ then
every conformal harmonic map $R\to\Omega$ from a parabolic conformal surface
$R$ is constant.

The minimal submanifolds in the theorem are not assumed to be complete or
proper. The hypothesis on the set of $m$-flat points of $b\Omega$ cannot be
omitted unless $3\leq m=n-1$. A counterexample is a halfspace $H$ of
$\mathbb{R}^{n}$, which contains parabolic minimal submanifolds of dimensions
$m\in\\{2,\ldots,n-2\\}$, or $m=2$ if $n=3$, contained in hyperplanes parallel
to $bH$. (Note that $\mathbb{R}^{m}$ with the flat metric is parabolic for
$m=2$, but is hyperbolic if $m\geq 3$, carrying the negative subharmonic
function $-1/|x|^{n-2}$.) When $m=2$, the minimal surfaces in the theorem are
allowed to have isolated branch points, and in this case the result is
essentially optimal since many open conformal surfaces of hyperbolic type
admit nonconstant bounded (and even complete) conformal harmonic maps to
$\mathbb{R}^{3}$. This holds in particular for any bordered conformal surface;
see [4, Chapter 7] for a survey of this topic.

A domain $\Omega\subset\mathbb{R}^{n}$ with $\mathscr{C}^{2}$ boundary whose
principal curvatures $\nu_{1}\leq\nu_{2}\leq\cdots\leq\nu_{n-1}$ from the
inner side satisfy $\nu_{1}+\nu_{2}+\cdots+\nu_{m}\geq 0$ at every point of
$b\Omega$ is said to be $m$-convex, and if $m=2$ it is also called minimally
convex; see [4, Definition 8.1.9] or Definition 3.1. By [4, Theorem 8.1.13]
this holds if and only if there exist a neighbourhood $U\subset\mathbb{R}^{n}$
of $b\Omega$ and a $\mathscr{C}^{2}$ function $\rho:U\to\mathbb{R}$ such that
$\Omega\cap U=\\{\rho<0\\}$, $d\rho\neq 0$ on $b\Omega\cap U=\\{\rho=0\\}$,
and

(2.1) $\mathrm{tr}_{\Lambda}\mathrm{Hess}_{\rho}(x)\geq 0\ \ \text{for every
point $x\in b\Omega$ and $m$-plane $\Lambda\subset T_{x}b\Omega$}.$

Here, $\mathrm{tr}_{\Lambda}\mathrm{Hess}_{\rho}(x)$ denotes the trace of the
restriction to $\Lambda$ of the Hessian form of $\rho$ at $x$. It was shown in
[15] that a bounded $m$-convex domain with $\mathscr{C}^{2}$ boundary in
$\mathbb{R}^{n}$ admits an $m$-plurisubharmonic defining function, i.e., one
satisfying condition (2.1) for every point $x\in\overline{\Omega}$ and
$m$-plane $\Lambda\subset\mathbb{R}^{n}$ (see (3.4)). The main new point shown
in this paper is that an unbounded domain $\Omega$ as in Theorem 2.1 admits a
defining function which is $m$-plurisubharmonic on $\overline{\Omega}$,
obtained by convexifying the signed distance function to $b\Omega$; see
Proposition 3.2. This is the key fact of independent interest used in the
proof of Theorem 2.1.

Note that $m$-convex domains play a major role in the theory of minimal
submanifolds. In particular, every bounded minimally convex domain contains
many properly immersed minimal surfaces parameterized by an arbitrary bordered
conformal surface; see [4, Theorems 8.3.1 and 8.3.4] for the orientable case
and [3, Theorem 6.9] for the nonorientable one. Such domains also form
barriers for minimal submanifolds; see Jorge and Tomi [25] and Gama et al.
[17], and the references therein. In [17] the authors proved the maximum
principle at infinity in a very general context including parabolic minimal
$m$-varifolds in $m$-convex domains. Nevertheless, I was unable to deduce
Theorem 2.1 from [17, Theorem 1.1] which holds under weaker hypotheses on
$b\Omega$ but seemingly stronger hypotheses on the minimal submanifolds.

The case of particular interest is when the boundary $M=b\Omega$ satisfies
$\nu_{1}+\cdots+\nu_{n-1}=0$, so it is a minimal hypersurface. If $M$ is not a
hyperplane then the set of its $(n-1)$-flat points has empty interior, and we
obtain the following corollary to Theorem 2.1.

######  Corollary 2.2.

Assume that $M$ is a properly embedded minimal hypersurface in
$\mathbb{R}^{n}$ for $n\geq 3$ such that for some $\epsilon>0$ every point in
the $\epsilon$-neighbourhood of $M$ has a unique nearest point in $M$. Then
every immersed parabolic minimal hypersurface $R\to\mathbb{R}^{n}$ intersects
$M$, unless $n=3$, $M$ is a plane in $\mathbb{R}^{3}$, and the image of $R$ is
contained in a plane parallel to $M$.

The condition in Theorem 2.1 and Corollary 2.1, that the $\mathscr{C}^{2}$
hypersurface $M=b\Omega$ admit a tubular neighbourhood of positive radius
$\epsilon>0$, is nontrivial when $\Omega$ is unbounded, which is the only case
of interest. If this holds, one says that $M$ has positive reach, a
terminology introduced by Federer [14, Sect. 4]. The reach of $M$ is the
supremum of the numbers $\epsilon>0$ having this property. A $\mathscr{C}^{2}$
hypersurface with positive reach has bounded principal curvatures (see the
proof of Proposition 3.2). The converse holds in several cases of interest.
Federer proved [14, Lemma 4.11] that a graph in $\mathbb{R}^{n}$ over a domain
in $\mathbb{R}^{n-1}$ with sufficiently nice boundary, such that the gradient
of the graphing function is Lipschitz, has positive reach. Every compact piece
of an embedded $\mathscr{C}^{2}$ hypersurface has positive reach.

If $M$ is a complete embedded minimal surface in $\mathbb{R}^{3}$ of finite
total Gaussian curvature, then every end of $M$ is a graph over the complement
of a disc in $\mathbb{R}^{2}$ whose graphing function has bounded second order
partial derivatives, and the ends are well separated (see Jorge and Meeks
[26]). Hence, every such surface has positive reach. It was shown by Meeks and
Rosenberg [38, Theorem 5.3] (see also [34, Corollary 2.6.6]) that a complete
embedded minimal surface with bounded curvature in $\mathbb{R}^{3}$ is proper
and has positive reach. This gives the following corollary to Theorem 2.1,
originally due to Pacelli Bessa, Jorge, and Pessoa [9, Theorem 1.1].

######  Corollary 2.3.

The image of a nonconstant conformal harmonic map $R\to\mathbb{R}^{3}$
(possibly with branch points) from a parabolic conformal surface intersects
every properly embedded nonflat minimal surface $M$ in $\mathbb{R}^{3}$ with
bounded Gaussian curvature.

######  Remark 2.4.

In [9, Theorem 1.1], Corollary 2.3 is proved under the weaker assumption that
$M$ is a complete nonflat immersed minimal surface in $\mathbb{R}^{3}$ of
bounded curvature. (Unlike in the case when $M$ is embedded, such a surface
need not be proper in $\mathbb{R}^{3}$ as shown by Andrade [6].) Their proof
for the immersed case is more involved. I thank G. Pacelli Bessa and L. F.
Pessoa for having pointed out their work after an earlier version of this
preprint was posted.

If one imposes stronger conditions on a minimal surface in a domain
$\Omega\subset\mathbb{R}^{3}$, then the conclusion of Corollary 2.3 holds
under weaker conditions on $b\Omega$. For example, if
$\Omega\subsetneq\mathbb{R}^{3}$ is a smoothly bounded minimally convex domain
and $R\subset\Omega$ is a complete immersed minimal surface of finite total
Gaussian curvature (hence proper in $\mathbb{R}^{3}$), then $R$ is a plane and
$\Omega$ is a slab or a halfspace (see [1, Theorem 1.16]). This also follows
from the results in [17].

Corollary 2.3 applies to every complete embedded nonflat minimal surface $M$
in $\mathbb{R}^{3}$ of finite total Gaussian curvature, as well as to many
minimal surfaces of infinite total curvature such as the standard helicoid,
the helicoids of positive genera constructed by Hoffman, Traizet, and White
[24], and Riemann’s minimal surfaces [35]. It was shown by Meeks, Pérez, and
Ros [36, Theorem 1] that every properly embedded minimal surface in
$\mathbb{R}^{3}$ of finite genus has bounded curvature, so Corollary 2.3 holds
for such surfaces. Furthermore, every periodic properly embedded minimal
surface in $\mathbb{R}^{3}$ whose fundamental domain is of finite topological
type satisfies Corollary 2.3 by results of Meeks and Rosenberg [37]; examples
include Scherk’s surfaces. We refer to Meeks and Pérez [34] for more
information on this topic.

Corollary 2.3 gives a partial negative answer to [13, Problem 1.16], asking
whether the complement of a nonflat properly embedded minimal surface in
$\mathbb{R}^{3}$ contains a minimal surface parameterized by $\mathbb{C}$. The
case of unbounded curvature (and hence necessarily of infinite genus by Meeks,
Pérez and Ros [36, Theorem 1]) remains open:

######  Problem 2.5.

Let $M$ be a properly embedded minimal surface in $\mathbb{R}^{3}$ with
unbounded curvature. Is there a nonconstant conformal harmonic map
$\mathbb{C}\to\mathbb{R}^{3}$ whose image avoids $M$?

##  3\. Proof of Theorem 2.1

We begin with preliminaries. Recall that a map
$f=(f_{1},\ldots,f_{n}):U\to\mathbb{R}^{n}$ from an open set
$U\subset\mathbb{C}$ is said to be conformal if it preserves angles at every
immersion point; equivalently, if $z=x+\mathfrak{i}y$ is a complex coordinate
on $U$ then the partial derivatives of $f$ satisfy $f_{x}\,\cdotp f_{y}=0$ and
$|f_{x}|=|f_{y}|$. Here, the dot denotes the Euclidean inner product and
$|\,\cdotp|$ the Euclidean length. These conditions imply that the rank of $f$
at any point is either two (an immersion point) or zero (a branch point). The
analogous definition applies to maps from any conformal surface in local
isothermal coordinates. The image of a nonconstant conformal harmonic map from
an open conformal surface is a minimal surface with isolated branch points;
conversely, every minimal surface with isolated branch points is of this form.
For background, see e.g. the monographs [4, 11, 39] and the surveys [2, 33,
34].

Given a subset $M$ of $\mathbb{R}^{n}$, we denote by $\mathrm{dist}(\cdotp,M)$
the Euclidean distance function to $M$:

$\mathrm{dist}(x,M)=\inf\\{|x-p|:p\in M\\},\quad x\in\mathbb{R}^{n}.$

For $p\in\mathbb{R}^{n}$ and $\epsilon>0$ we let
$\mathbb{B}(p,\epsilon)=\\{x\in\mathbb{R}^{n}:|x-p|<\epsilon\\}$ and

(3.1) $V_{\epsilon}(M)=\bigcup_{p\in
M}\mathbb{B}(p,\epsilon)=\left\\{x\in\mathbb{R}^{n}:\mathrm{dist}(x,M)<\epsilon\right\\}\\!.$

Let $M$ be a properly embedded connected hypersurface in $\mathbb{R}^{n}$.
Such $M$ is orientable and its complement $\mathbb{R}^{n}\setminus M=M^{+}\cup
M^{-}$ consists of a pair of connected domains. The signed distance function
to $M$ is defined by

(3.2) $\delta_{M}(x)=\begin{cases}\phantom{-}\mathrm{dist}(x,M),&\text{if}\
x\in M_{+}\cup M;\\\ -\mathrm{dist}(x,M),&\text{if}\ x\in M_{-}.\end{cases}$

If $V\subset\mathbb{R}^{n}$ is an open neighbourhood of $M$ such that every
point $x\in V$ has a unique nearest point $p=\xi(x)\in M$, and if $M$ is of
class $\mathscr{C}^{r}$ for some $r\in\\{2,3,\ldots,\infty,\omega\\}$, then
$\delta_{M}$ is also of class $\mathscr{C}^{r}$ on $V$ (see Gilbarg and
Trudinger [18, Lemma 14.16] or Krantz and Park [29]).

We recall the following notion; see [4, Sections 8.1.4–8.1.5].

######  Definition 3.1.

An oriented embedded $\mathscr{C}^{2}$ hypersurface $M\subset\mathbb{R}^{n}$
is $m$-convex for some $m\in\\{1,2,\ldots,n-1\\}$ if its principal curvatures
$\nu_{1}\leq\nu_{2}\leq\cdots\leq\nu_{n-1}$ satisfy

(3.3) $\nu_{1}(p)+\nu_{2}(p)+\cdots+\nu_{m}(p)\geq 0\ \ \text{for all $p\in
M$}.$

The hypersurface $M$ is strongly $m$-convex at $p\in M$ if strong inequality
holds in (3.3). A point $p\in M$ is $m$-flat if $\nu_{j}(p)=0$ for
$j=1,\ldots,m$. We say that $M$ is not $m$-flat at infinity if the set
$\\{p\in M:\nu_{j}(p)=0\ \text{for}\ j=1,\ldots,m\\}$ has bounded relative
interior in $M$.

Given a $\mathscr{C}^{2}$ function $\rho:\Omega\to\mathbb{R}$ on a domain
$\Omega\subset\mathbb{R}^{n}$, we denote by $\mathrm{Hess}_{\rho}(x)$ the
Hessian of $\rho$ at the point $x\in\Omega$, i.e., the quadratic form on
$T_{x}\mathbb{R}^{n}=\mathbb{R}^{n}$ represented by the matrix
$\big{(}\frac{\partial^{2}\rho}{\partial x_{i}\partial x_{j}}(x)\big{)}$ of
second order partial derivatives of $\rho$ at $x$. For $1\leq m<n$ let
$G_{m}(\mathbb{R}^{n})$ denote the Grassman manifold of $m$-planes in
$\mathbb{R}^{n}$. Given $\Lambda\in G_{m}(\mathbb{R}^{n})$, we denote by
$\mathrm{tr}_{\Lambda}\mathrm{Hess}_{\rho}(x)\in\mathbb{R}$ the trace of the
restriction of the Hessian $\mathrm{Hess}_{\rho}(x)$ to $\Lambda$. A
$\mathscr{C}^{2}$ function $\rho:\Omega\to\mathbb{R}$ is said to be
$m$-plurisubharmonic if

(3.4) $\mathrm{tr}_{\Lambda}\mathrm{Hess}_{\rho}(x)\geq 0\ \ \text{for all
$x\in\Omega$ and $\Lambda\in G_{m}(\mathbb{R}^{n})$}.$

If $\lambda_{1}(x)\leq\lambda_{2}(x)\leq\cdots\leq\lambda_{n}(x)$ are the
eigenvalues of $\mathrm{Hess}_{\rho}(x)$, then (3.4) is equivalent to

$\lambda_{1}(x)+\cdots+\lambda_{m}(x)\geq 0\ \ \text{for every}\ x\in\Omega.$

In fact,
$\lambda_{1}(x)+\cdots+\lambda_{m}(x)=\inf_{\Lambda}\mathrm{tr}_{\Lambda}\mathrm{Hess}_{\rho}(x).$
(See [4, Sect. 8.1] or [22].) The main point is that a function
$\rho:\Omega\to\mathbb{R}$ is $m$-plurisubharmonic if and only if its
restriction to every $m$-dimensional minimal submanifold in $\Omega$ is a
subharmonic function on the submanifold (see [4, Proposition 8.1.2]). The
notion of an $m$-plurisubarmonic function extends to upper semicontinuous
functions by asking that for any $m$-dimensional affine subspace $L$ of
$\mathbb{R}^{n}$, the restriction of the function to $L\cap\Omega$ is a
subharmonic function (see [4, Definition 8.1.1]). Such a function can be
approximated from above on any relatively compact subdomain of $\Omega$ by
smooth $m$-plurisubarmonic functions (see [4, Proposition 8.1.6]). In this
paper, we shall use continuous $m$-plurisubharmonic functions obtained by
taking the maximum of a smooth $m$-plurisubharmonic function and a constant,
and in this case, smoothing can be obtained globally in a simple way; see the
last part of the proof of Proposition 3.2.

Theorem 2.1 is a consequence of the following proposition of independent
interest.

######  Proposition 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain with $\mathscr{C}^{r}$ boundary
for some $r\geq 2$ satisfying the assumptions in Theorem 2.1. There is a
function $\rho:\overline{\Omega}\to(-\infty,0]$ of class
$\mathscr{C}^{r}(\overline{\Omega})$ which is $m$-plurisubharmonic on
$\Omega$, it vanishes on $M=b\Omega$, it satisfies $d\rho\neq 0$ on
$\rho^{-1}((-1,0])$, and for every $t\in(-1,0)$ the level set
$M_{t}=\\{\rho=t\\}$ coincides with a level set of the signed distance
function $\delta_{M}$ (3.2), $M_{t}$ is $m$-convex, it is strongly $m$-convex
at every $m$-nonflat point, and $M_{t}$ is not $m$-flat near infinity.

We point out that, for the proof of Theorem 2.1, it suffices to find a
continuous $m$-plurisubharmonic function
$\rho:\overline{\Omega}\to(-\infty,0]$ which is of class $\mathscr{C}^{2}$ on
an interior collar around $M=b\Omega$ and has the other stated properties.
Such a function is given by (3.18).

###### Proof.

By Gilbarg and Trudinger [18, Lemma 14.16] or Krantz and Parks [29], the
conditions on $M=b\Omega$ imply that the signed distance function
$\delta=\delta_{M}$ to $M$ (3.2) is of class $\mathscr{C}^{r}$ on the
$\epsilon$-neighbourhood $V_{\epsilon}(M)$ of $M$ (see (3.1)). We choose the
sign so that $\delta<0$ on

(3.5) $C_{\epsilon}:=\Omega\cap
V_{\epsilon}(M)=\\{x\in\Omega:\mathrm{dist}(x,M)<\epsilon\\}.$

We recall some further properties of $\delta$, referring to Bellettini [7,
Theorem 1.18, p. 14] and Gilbarg and Trudinger [18, Section 14.6]. There is a
projection $\xi:V_{\epsilon}(M)\to M$ of class $\mathscr{C}^{r-1}$ such that
for every $x\in V_{\epsilon}(M)$ the point $p=\xi(x)\in M$ is the unique
nearest point to $x$ in $M$. The gradient $\nabla\delta$ has constant norm
$|\nabla\delta|=1$ on $V_{\epsilon}(M)$, and it has constant value on the
intersection of $V_{\epsilon}(M)$ with the normal line
$N_{p}=p+\mathbb{R}\,\cdotp\nabla\delta(p)$ at $p\in b\Omega=M$. There an
orthonormal basis $(v_{1},\ldots,v_{n-1},v_{n}=\nabla\delta(p))$ of
$\mathbb{R}^{n}$ such that the vectors $v_{1},\cdots,v_{n-1}$ span $T_{p}M$,
and in this basis the (symmetric) matrix $A(p)$ of $\mathrm{Hess}_{\delta}(p)$
is diagonal:

(3.6) $A(p)v_{j}=\nu_{j}v_{j}\ \ \text{for $j=1,\ldots,n-1$};\quad
A(p)v_{n}=0,$

where the numbers $\nu_{1}\leq\nu_{2}\leq\cdots\leq\nu_{n-1}$ are the
principal normal curvatures of $M$ at $p$ from the interior side. For any
point $x\in N_{p}\cap V_{\epsilon}(M)$ the same basis $(v_{1},\ldots,v_{n})$
diagonalizes $\mathrm{Hess}_{\delta}(x)$, with the corresponding eigenvalues

(3.7) $\nu_{j}(x)=\frac{\nu_{j}}{1+\delta(x)\nu_{j}}\ \ \text{for
$j=1,\ldots,n-1$};\quad A(p)v_{n}=0.$

If $\nu_{j}>0$ for some $j$ then the above shows that $1+\delta(x)\nu_{j}>0$
for all $x\in C_{\epsilon}\cap N_{p}$. Since $\delta(x)$ assumes all values in
$(-\epsilon,0)$ on the set $C_{\epsilon}$ (3.5), this implies
$1-\epsilon\nu_{j}\geq 0$ and hence $\nu_{j}\leq 1/\epsilon$ for all such $j$.
From this and the hypothesis $\nu_{1}+\nu_{2}+\cdots+\nu_{m}\geq 0$ it also
follows that $\nu_{j}\geq-(m-1)/\epsilon$ for every $j=1,\ldots,n-1$.
Summarising, we have that

(3.8) $-(m-1)/\epsilon\leq\nu_{j}(p)\leq 1/\epsilon\ \ \text{for all
$j=1,\ldots,n-1$ and $p\in M$}.$

That is, the hypersurface $M=b\Omega$ has bounded principal curvatures. The
worst case estimate in the first inequality in (3.8) occurs only when
$\nu_{1}=-(m-1)/\epsilon$ and $\nu_{j}=1/\epsilon$ for $j=2,\ldots,m$, so we
also conclude that

(3.9) $\sum_{\nu_{j}(p)\leq 0}\nu_{j}(p)\geq-\frac{m-1}{\epsilon}\ \ \text{for
all $p\in M$.}$

Consider the family of domains

$\Omega_{t}=\\{x\in C_{\epsilon}:\delta(x)<t\\}\cup(\Omega\setminus
C_{\epsilon})\quad\text{for $t\in(-\epsilon,0]$.}$

As $t$ increases towards $0$, the domains $\Omega_{t}$ increase to
$\Omega_{0}=\Omega$. The tangent space to the hypersurface
$b\Omega_{t}=\\{\delta=t\\}$ at the point $x=p+t\nabla\delta(p)$ is spanned by
the same vectors $v_{1},\ldots,v_{n-1}$ as $T_{p}M$ (see (3.6)), and the
numbers $\nu_{j}(x)$ in (3.7) for $j=1,\ldots,n-1$ are the principal
curvatures of $b\Omega_{t}$ at $x$. As $\delta(x)$ decreases (we move away
from $M$), each of the curvatures $\nu_{j}(x)$ strictly increases unless
$\nu_{j}(p)=0$ (in whihc case $\nu_{j}(x)=0$), and it does not change the
sign. This implies that for every $x=p+t\nabla\delta(p)\in b\Omega_{t}$ with
$t\in(-\epsilon,0)$ we have

(3.10) $\displaystyle\nu_{1}(x)\leq\nu_{2}(x)\leq\cdots\leq\nu_{n-1}(x),\ \
\nu_{j}(x)\geq\nu_{j}(p)\ \ \text{for $j=1,\ldots,n-1$},\ \ \text{and}$ (3.11)
$\displaystyle
H(x):=\nu_{1}(x)+\nu_{2}(x)+\cdots+\nu_{m}(x)\geq\nu_{1}(p)+\nu_{2}(p)+\cdots+\nu_{m}(p)\geq
0,$

where the first inequality in (3.11) is equality if and only if the point
$p\in M$ is $m$-flat (see Definition 3.1). Indeed, the function $H$ increases
as we move away from $M=b\Omega$ into $\Omega$, and it vanishes on $N_{p}\cap
C_{\epsilon}$ if and only if the point $p\in M$ is $m$-flat. We conclude that
for $t\in(-\epsilon,0)$ the hypersurface $M_{t}=b\Omega_{t}$ is $m$-convex,
and it is strongly $m$-convex at $x=p+t\nabla\delta(p)\in C_{\epsilon}$ if and
only if the point $p=\xi(x)\in M$ is not $m$-flat. In particular, since $M$ is
not $m$-flat near infinity by theassumption, $M_{t}$ is not $m$-flat near
infinity for any $t\in(-\epsilon,0)$.

From (3.7)–(3.10) it follows that for every $x\in C_{\epsilon/2}$ (see (3.5))
we have that

(3.12)
$\displaystyle\displaystyle{-\frac{m-1}{\epsilon}\leq\nu_{j}(x)\leq\frac{2}{\epsilon}\
\ \text{for $j=1,\ldots,n-1$, and}}$ (3.13)
$\displaystyle\displaystyle{\sum_{\nu_{j}(x)\leq
0}\nu_{j}(x)\geq-\frac{m-1}{\epsilon}.}$

Despite the estimate (3.11), the function $\delta$ need not be
$m$-plurisubharmonic on $C_{\epsilon}$ due the zero eigenvalue of
$\mathrm{Hess}\,\delta$ in the normal direction determined by $\nabla\delta$.
However, we now show that a suitable convexification of $\delta$ on a collar
$C_{\epsilon_{0}}\subset\Omega$ in (3.5) for some $0<\epsilon_{0}<\epsilon/2$
extends to an $m$-plurisubharmonic function $\rho:\Omega\to(-\infty,0)$ such
that every level set

$M_{t}=\\{\rho=t\\}\ \ \text{for $t\in(-\epsilon_{0},0)$}$

coincides with a level set $\\{\delta=\tau\\}$ for some
$\tau=\tau(t)\in(-\epsilon,0)$. The argument is similar to [15, proof of
Theorem 1.1], but we are now dealing with an unbounded domain $\Omega$, and we
obtain uniform estimates of the quantities involved.

Choose a number $\alpha>(m-1)/\epsilon$. Let $h:\mathbb{R}\to\mathbb{R}$ be a
smooth, convex, increasing function with $h(0)=0$, $\dot{h}(0)=1$, and
$\ddot{h}(0)=\alpha$. Then,

(3.14) $h(t)<0\ \ \text{and}\ \ 0\leq\dot{h}(t)<1\ \ \text{for}\ \
-\infty<t<0.$

Choose numbers $\epsilon_{0},\epsilon^{\prime}_{0}$ with
$0<\epsilon_{0}<\epsilon^{\prime}_{0}<\epsilon/2$ such that

(3.15) $\ddot{h}(t)>\frac{m-1}{\epsilon}\ \ \text{for
$-\epsilon^{\prime}_{0}\leq t\leq 0$}.$

Consider the function $h\circ\delta:C_{\epsilon}\to(-\infty,0)$. We have that

(3.16)
$\mathrm{Hess}_{h\circ\delta}=(\dot{h}\circ\delta)\mathrm{Hess}_{\delta}+(\ddot{h}\circ\delta)\nabla\delta\,\cdotp(\nabla\delta)^{T}.$

(Here, $(\nabla\delta)^{T}$ is the transpose of $\nabla\delta$, and
$\nabla\delta\,\cdotp(\nabla\delta)^{T}$ is an $n\times n$ matrix.) Recall
that for $p\in M$ the orthonormal vectors $v_{1},\cdots
v_{n-1},v_{n}=\nabla\delta(p)$ diagonalize $\mathrm{Hess}_{\delta}(p)$, where
$T_{p}M=\mathrm{span}\\{v_{1},v_{2},\ldots,v_{n-1}\\}$. Note that $T_{p}M$
lies in the kernel of the matrix $\nabla\delta\,\cdotp(\nabla\delta)^{T}$
while $v_{n}$ is an eigenvector with eigenvalue $1$. The same basis then
diagonalizes $\mathrm{Hess}_{h\circ\delta}(x)$ at every point
$x=p+\delta(x)v_{n}\in N_{p}\cap C_{\epsilon}$, the eigenvalues corresponding
to $v_{1},\ldots,v_{n-1}$ get multiplied by $\dot{h}(\delta(x))\geq 0$, and
the eigenvalue in the normal direction $v_{n}$ equals $\ddot{h}(\delta(x))$.
Summarising, the eigenvalues of $\mathrm{Hess}_{h\circ\delta}(x)$ at any point
$x\in C_{\epsilon}$ equal

(3.17)
$\dot{h}(\delta(x))\nu_{1}(x),\ldots,\dot{h}(\delta(x))\nu_{n-1}(x),\quad\ddot{h}(\delta(x)).$

If $x\in C_{\epsilon^{\prime}_{0}}$ then $-\epsilon^{\prime}_{0}<\delta(x)<0$,
and hence by (3.13), (3.14) and (3.15) we have that

$\sum_{\nu_{j}(x)\leq 0}\dot{h}(\delta(x))\nu_{j}(x)+\ddot{h}(\delta(x))>0.$

Together with (3.10) and (3.11) it follows that the sum of any $m$ numbers in
the list (3.17) is nonnegative at every point $x\in
C_{\epsilon^{\prime}_{0}}$, i.e., the function $h\circ\delta$ is
$m$-plurisubharmonic on $C_{\epsilon^{\prime}_{0}}$. Hence, the continuous
function $\rho_{0}:\Omega\to[h(-\epsilon_{0}),0)$ given by

(3.18) $\rho_{0}(x)=\begin{cases}h(\delta(x)),&\text{if}\ x\in
C_{\epsilon_{0}};\\\ h(-\epsilon_{0}),&\text{if}\ x\in\Omega\setminus
C_{\epsilon_{0}}\end{cases}$

is $m$-plurisubharmonic. Indeed, near $bC_{\epsilon_{0}}$ we have that
$\rho_{0}=\max\\{h\circ\delta,h(-\epsilon_{0})\\}$, and the maximum of two
$m$-plurisubharmonic functions is $m$-plurisubharmonic (see [21, Sect. 6]).

To get a smooth $m$-plurisubharmonic function $\rho\in\mathscr{C}^{r}(\Omega)$
which agrees with $\rho_{0}$ on a smaller collar $C_{\epsilon_{1}}\subset
C_{\epsilon_{0}}$, we choose numbers
$0<\epsilon_{1}<\epsilon_{2}<\epsilon_{0}$, pick a smooth increasing convex
function $\chi:\mathbb{R}\to\mathbb{R}$ such that $\chi(t)$ is a negative
constant for $t\leq h(-\epsilon_{2})$ and $\chi(t)=t$ for $t\geq
h(-\epsilon_{1})$, and set $\rho=\chi\circ\rho_{0}$. Clearly, $\rho$ is well-
defined on $\overline{\Omega}$ and of class $\mathscr{C}^{r}$, and it is
constant on $\Omega\setminus C_{\epsilon_{2}}$. Using the formula (3.16) for
points in $C_{\epsilon_{0}}$ gives

$\mathrm{Hess}_{\chi\circ\rho_{0}}=(\dot{\chi}\circ\rho_{0})\mathrm{Hess}_{\rho_{0}}+(\ddot{\chi}\circ\rho)\nabla\rho_{0}\,\cdotp(\nabla\rho_{0})^{T}.$

Since $\dot{\chi}\circ\rho_{0}\geq 0$ and $\ddot{\chi}\circ\rho\geq 0$,
$\rho_{0}$ is $m$-plurisubharmonic on $C_{\epsilon_{0}}$, and the matrix
$\nabla\rho_{0}\,\cdotp(\nabla\rho_{0})^{T}$ is nonnegative definite, we infer
that $\rho=\chi\circ\rho_{0}$ is of class $\mathscr{C}^{r}$ and
$m$-plurisubharmonic on $\overline{\Omega}$. Clearly, $\rho$ satisfies the
conclusion of the proposition for all $t\in(h(-\epsilon_{1}),0)$. For such
$t$, we have $\chi(t)=t$ and hence the hypersurface
$M_{t}=\\{\rho=t\\}=\\{h\circ\delta=t\\}$ equals

(3.19) $M_{t}=\\{\delta=\tau\\}\ \ \text{for
$\tau=h^{-1}(t)\in(-\epsilon_{1},0)$}.$

Replacing $\rho$ by $c\rho$ with $c=-1/h(-\epsilon_{1})>0$, this holds for
$t\in(-1,0)$. ∎

###### Proof of Theorem 2.1.

Since an $m$-convex domain is also $k$-convex for any $m<k\leq n-1$, and every
$k$-flat point is also $m$-flat, it suffices to show that the domain $\Omega$
in the theorem does not contain any parabolic minimal surfaces of dimension
$m$.

Let $\rho:\Omega\to(-\infty,0)$ be an $m$-plurisubharmonic function given by
Proposition 3.2. If $f:R\to\Omega$ is an immersed minimal submanifold of
dimension $m$, then $\rho\circ f$ is a negative function on $R$ which is
subharmonic in the Riemannian metric induced by $f$ from the Euclidean metric
on $\mathbb{R}^{n}$. If in addition this minimal surface is connected and
parabolic, then this function is constant. Thus, either $f(R)\subset
M_{t}=\\{\rho=t\\}$ for some $t\in(-1,0)$, or $f(R)\subset\Omega\setminus
C_{\epsilon_{1}}$ where $\epsilon_{1}>0$ is as in (3.19) and
$C_{\epsilon_{1}}$ is given by (3.5).

Assume that the first possibility holds. The minimal $m$-dimensional
submanifold $f(R)\subset M_{t}$ is clearly contained in the set of points
where the hypersurface $M_{t}$ is not strongly $m$-convex. By Proposition 3.2,
$M_{t}$ is $m$-flat at every point where it fails to be strongly $m$-convex,
and the set of its $m$-flat points has bounded interior. Since $f(R)$ is
parabolic, it cannot be contained in a bounded set, so this case is
impossible.

Therefore, $f(R)\subset\Omega\setminus C_{\epsilon_{1}}$, so the distance
between $f(R)$ and $b\Omega$ is at least $\epsilon_{1}>0$. If we translate
$\Omega$ for the distance $\epsilon_{1}/2$ in any direction, the same argument
(with the same constants) applies to the translated domain $\Omega^{\prime}$
and its boundary $b\Omega^{\prime}$. Since $f(R)\subset\Omega^{\prime}$, we
infer that $\mathrm{dist}(f(R),b\Omega^{\prime})\geq\epsilon_{1}$. Since this
holds for every translate $\Omega^{\prime}$ of $\Omega$ for $\epsilon_{1}/2$,
we conclude that $\mathrm{dist}(f(R),b\Omega)\geq 3\epsilon_{1}/2$. Repeating
this argument shows that the distance from $f(R)$ to $b\Omega$ must be
infinite, a contradiction.

If $m=2$, the same argument shows that there are no nonconstant conformal
harmonic maps $f:R\to\Omega$ (possibly with branch points) from any parabolic
conformal surface $R$. (In this case, parabolicity does not depend on $f$ but
only on the conformal structure of $R$.) ∎

######  Remark 3.3 (Concerning the paper [15]).

By [15, Theorem 1.1], every bounded domain $\Omega$ in $\mathbb{R}^{n}$
$(n\geq 3)$, whose boundary is of class $\mathscr{C}^{r,\alpha}$ for some
$r\geq 2$ and $0<\alpha\leq 1$ and is $m$-convex for some
$m\in\\{1,\ldots,n-1\\}$, admits an $m$-plurisubharmonic defining function of
class $\mathscr{C}^{r,\alpha}$. The proof in the cited paper refers to the
result of Li and Nirenberg [30] saying that the signed distance function to a
hypersurface of class $\mathscr{C}^{r,\alpha}$ for such $(r,\alpha)$ is of the
same class $\mathscr{C}^{r,\alpha}$ near the hypersurface. By using the result
of Gilbarg and Trudinger [18, Lemma 14.16] and Krantz and Parks [29], we see
that [15, Theorem 1.1] also holds in smoothness classes $\mathscr{C}^{r}$ for
$r=2,3,\ldots,\infty,\omega$.

The second remark is that the last statement in [15, Theorem 1.1], concerning
strongly $m$-convex domains, is one of several special cases of a result by
Harvey and Lawson [20, Theorem 5.12]. I wish to thank the authors for the
relevant communication.

##  4\. Weakly hyperbolic domains

As mentioned in the introduction, the second motivation for this paper comes
from the recently introduced hyperbolicity theory for domains in
$\mathbb{R}^{n}$ $(n\geq 3)$ in terms of minimal surfaces. We recall the
background.

In 2021, Forstnerič and Kalaj [16] defined on any domain $\Omega$ in
$\mathbb{R}^{n}$ for $n\geq 3$ a Finsler pseudometric
$g_{\Omega}:T\Omega=\Omega\times\mathbb{R}^{n}\to\mathbb{R}_{+}=[0,+\infty)$,
called the minimal pseudometric, and the associated pseudodistance
$\mathrm{dist}_{\Omega}:\Omega\times\Omega\to\mathbb{R}_{+}$ obtained by
integrating $g_{\Omega}$, by using conformal harmonic discs
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}\to\Omega$, in the same way as the
Kobayashi–Royden pseudometric and pseudodistance are defined on any complex
manifold by using holomorphic discs; see [27, 28, 41]. We recall the
definition. Let $\mathrm{CH}(\mathbb{D},\Omega)$ denote the space of conformal
harmonic discs $f:\mathbb{D}\to\Omega$, possibly with branch points. Denote by
$z=x+\mathfrak{i}y$ the complex coordinate on $\mathbb{D}$ and by $f_{x}$ the
partial derivative of $f$ on $x$. The minimal pseudometric at $p\in\Omega$ on
a vector $v\in\mathbb{R}^{n}$ is defined by

$g_{\Omega}(p,v)=\inf\bigl{\\{}1/r>0:\exists
f\in\mathrm{CH}(\mathbb{D},\Omega),\ f(0)=p,\ f_{x}(0)=rv\bigr{\\}}\geq 0.$

It follows from definitions that $\mathrm{dist}_{\Omega}$ is the biggest
pseudometric on $\Omega$ such that every conformal harmonic map $R\to\Omega$
from a conformal surface $R$ is distance decreasing with respect to the
Poincaré pseudometric on $R$. Thus, $\mathrm{dist}_{\Omega}$ describes the
fastest possible rate of growth of hyperbolic minimal surfaces in $\Omega$. A
domain $\Omega$ is said to be hyperbolic if $\mathrm{dist}_{\Omega}$ is a
distance function, and complete hyperbolic if
$(\Omega,\mathrm{dist}_{\Omega})$ is a complete metric space. Every bounded
domain is hyperbolic but not necessarily complete hyperbolic. On the unit ball
of $\mathbb{R}^{n}$, the minimal metric coincides with the
Beltrami–Cayley–Klein metric, one of the classical models of hyperbolic
geometry; see [16]. We refer to [12] for the basic hyperbolicity theory for
domains in Euclidean spaces.

We now introduce the following weaker notion of hyperbolicity, which is of
interest on unbounded domains where hyperbolicity remains poorly understood.

######  Definition 4.1.

A domain $\Omega$ in $\mathbb{R}^{n}$, $n\geq 3$, is weakly hyperbolic (for
minimal surfaces) if every conformal harmonic map $\mathbb{C}\to\Omega$ is
constant.

Weak hyperbolicity is an analogue of Brody hyperbolicity in complex analysis:
a complex manifold $X$ is said to be Brody hyperbolic if every holomorphic map
$\mathbb{C}\to X$ is constant. Every Kobayashi hyperbolic manifold is clearly
also Brody hyperbolic; the converse holds on compact complex manifolds (see
Brody [10]) but it fails in general (see [10, p. 219]).

Likewise, every hyperbolic domain in $\mathbb{R}^{n}$ is weakly hyperbolic but
the converse fails in general. For the first claim, assume that
$\Omega\subset\mathbb{R}^{n}$ is not weakly hyperbolic. Pick a nonconstant
conformal harmonic map $f:\mathbb{C}\to\Omega$ and a point $a\in\mathbb{C}$
where $df_{a}\neq 0$. The conformal harmonic discs $f_{r}:\mathbb{D}\to\Omega$
given by $f_{r}(z)=f(a+rz)$ for $r>0$ satisfy $f_{r}(0)=f(a)$ and
$d(f_{r})_{0}=rdf_{a}$. Letting $r\to+\infty$ we get $g_{\Omega}(f(a),v)=0$
for all $v\in df_{a}(\mathbb{C})$, so $\Omega$ is not hyperbolic. The failure
of the converse implication is given by part (b) of the following proposition.
Part (c) shows that a weakly hyperbolic domain may contain parabolic minimal
surfaces whose universal conformal covering surface is the disc.

######  Proposition 4.2.

  1. (a)

There is a nonhyperbolic domain in $\mathbb{R}^{3}$ which does not contain any
parabolic minimal surfaces.

  2. (b)

In particular, there is a weakly hyperbolic domain in $\mathbb{R}^{3}$ which
is not hyperbolic.

  3. (c)

For every parabolic domain $D\subsetneq\mathbb{R}^{2}=\mathbb{C}$ which omits
at least two points of $\mathbb{C}$ there is a weakly hyperbolic but
nonhyperbolic domain $\Omega_{D}\subset\mathbb{R}^{3}$ containing
$D\times\\{0\\}$ as a proper minimal surface.

###### Proof.

Given a domain $D\subset\mathbb{R}^{2}$, we define the domain
$\Omega_{D}\subset\mathbb{R}^{3}$ by

$\Omega_{D}=\bigl{\\{}(x,y,z)\in\mathbb{R}^{3}:|z|<1,\ z^{2}(x^{2}+y^{2})<1,\
(x,y)\in D\ \text{if}\ z=0\bigr{\\}}.$

We claim that the following assertions hold.

  1. (i)

The minimal distance in $\Omega_{D}$ between any pair of points $p,q\in
D\times\\{0\\}$ vanishes.

  2. (ii)

The image of every nonconstant conformal harmonic map $R\to\Omega_{D}$ from a
parabolic conformal surface $R$ is contained in $D\times\\{0\\}$.

To prove the first claim (i), assume that $p=(a,b,0)$ and $q=(c,d,0)$. Set
$p_{k}=(a,b,1/k)$ and $q_{k}=(c,d,1/k)$ for $k=2,3,\ldots$. It is obvious that
$\lim_{k\to\infty}\mathrm{dist}_{\Omega_{D}}(p,p_{k})=0$ and
$\lim_{k\to\infty}\mathrm{dist}_{\Omega_{D}}(q,q_{k})=0$. The sequence of
linear discs $\\{(x,y,1/k):x^{2}+y^{2}<k^{2}\\}\subset\Omega_{D}$ shows that
$\lim_{k\to\infty}\mathrm{dist}_{\Omega_{D}}(p_{k},q_{k})=0$. Hence,
$\mathrm{dist}_{\Omega_{D}}(p,q)=0$.

To prove the second claim (ii), assume that $R$ is a parabolic conformal
surface and $f=(f_{1},f_{2},f_{3}):R\to\Omega_{D}$ is a nonconstant conformal
harmonic map. We have that $|f_{3}|<1$, so $f_{3}=c$ is constant. If $c\neq 0$
then $f_{1}^{2}+f_{2}^{2}<1/c^{2}<+\infty$, so $f_{1}$ and $f_{2}$ are
constant as well, a contradiction. Thus, $c=0$ and hence $f(R)\subset
D\times\\{0\\}$.

Taking $D$ to be a bounded domain in $\mathbb{R}^{2}$, property (ii) implies
that $\Omega_{D}$ does not contain any parabolic minimal surfaces. Since
$\Omega_{D}$ is nonhyperbolic by property (i), such a domain satisfies
condition (a) in the proposition. Part (b) is an immediate consequence.

Let $D\subset\mathbb{C}$ be a parabolic domain which omits at least two points
of $\mathbb{C}$. By (ii) the image of every nonconstant conformal harmonic map
$\mathbb{C}\to\Omega_{D}$ is contained in $D\times\\{0\\}$, which contradicts
Picard’s theorem. Thus, $\Omega_{D}$ is weakly hyperbolic but it contains the
parabolic minimal surface $D\times\\{0\\}$. This proves part (c) of the
proposition. ∎

######  Problem 4.3.

Is there a hyperbolic domain $\Omega\subset\mathbb{R}^{3}$ containing a
parabolic minimal surface? In particular, if
$D\subset\mathbb{R}^{2}=\mathbb{C}$ is a parabolic domain which omits at least
two points of $\mathbb{C}$, is the domain
$D\times(-\epsilon,\epsilon)\subset\mathbb{R}^{3}$ hyperbolic for some (or
all) $\epsilon>0$?

Note that the domain $(\mathbb{C}\setminus\\{0,1\\})\times\mathbb{D}$ in
$\mathbb{C}^{2}$ is Kobayashi hyperbolic. The difference between the two cases
is that the projection of a complex curve to any factor in a product of
complex manifolds is holomorphic, while the projection of a conformal harmonic
map is harmonic but not conformal in general. It is easily seen that
$\mathbb{C}\setminus\\{0,1\\}$ contains many harmonic images of $\mathbb{C}$
which are conformal at some point, but not everywhere.

The situation is much simpler for convex domains as shown by the following
proposition.

######  Proposition 4.4.

For a convex domain $\Omega$ in $\mathbb{R}^{n}$, $n\geq 3$, the following are
equivalent.

  1. (a)

$\Omega$ is complete hyperbolic.

  2. (b)

$\Omega$ is hyperbolic.

  3. (c)

$\Omega$ is weakly hyperbolic.

  4. (d)

$\Omega$ does not contain any affine $2$-plane.

  5. (e)

$\Omega$ does not contain any parabolic minimal surface.

###### Proof.

The implications $(a)\Rightarrow(b)\Rightarrow(c)\Rightarrow(d)$ and
$(e)\Rightarrow(d)$ are trivial. The implication $(d)\Rightarrow(a)$ was
proved in [12, Theorem 5.1], which establishes the equivalence of properties
$(a)$–$(d)$. It remains to prove $(d)\Rightarrow(e)$. It was shown in [12,
proof of Theorem 5.1] that a convex domain $\Omega$ in $\mathbb{R}^{n}$
satisfying condition $(d)$ is contained in the intersection of $n-1$
halfspaces $H_{j}=\\{\ell_{j}<c_{j}\\}$ $(j=1,\ldots,n-1)$, where
$\ell_{1},\ldots,\ell_{n-1}:\mathbb{R}^{n}\to\mathbb{R}$ are linearly
independent linear functionals and $c_{j}$ are constants. If $f:R\to\Omega$ is
a conformal harmonic map from a connected parabolic conformal surface $R$ then
$\ell_{j}\circ f:R\to(-\infty,c_{j})$ is a bounded from above harmonic
function for every $j=1,\ldots,n-1$, hence constant. Thus, $f(R)$ lies in a
real line and hence $f$ is constant. ∎

A property of a domain directly opposite to weak or strong hyperbolicity is
flexibility for minimal surfaces, introduced in [13, Definition 1.1]. A domain
$\Omega\subset\mathbb{R}^{n}$ for $n\geq 3$ is said to be flexible if, given
an open conformal surface $M$, a compact set $K\subset M$ whose complement has
no relatively compact connected components, and a conformal harmonic immersion
$f:U\to\Omega$ from an open neighbourhood of $K$, we can approximate $f$
uniformly on $K$ by conformal harmonic immersions $M\to\Omega$. A flexible
domain admits many conformal harmonic images of $\mathbb{C}$, so it is not
weakly hyperbolic. In particular, the domains in Theorem 2.1 for $m=2$ are not
flexible. A halfspace is neither (weakly) hyperbolic nor flexible.

It is not surprising that the situation regarding Corollary 2.3 is quite
different in $\mathbb{R}^{4}$. As shown in [13, Example 1.9], a domain in
$\mathbb{R}^{4}$ with coordinates $(x_{1},x_{2},x_{3},x_{4})$, given by

$\Omega=\\{x_{4}>-a|x_{2}|+b|x_{3}|\ \ \ \text{for some $a>0$ and
$b\in\mathbb{R}$}\\},$

is flexible. Taking $b>0$, the complementary domain
$\Omega^{\prime}=\mathbb{R}^{4}\setminus\overline{\Omega}$ is of the same type
with the reversed roles of $x_{2}$ and $x_{3}$, and with $x_{4}$ replaced by
$-x_{4}$, so it is also flexible. By Alarcón and López [5] each of these
domains contains a properly immersed conformal minimal surface parameterised
by an arbitrary open Riemann surface. (In fact, their result holds for any
concave wedge in $\mathbb{R}^{3}$ obtained by intersecting $\Omega$ with the
hyperplane $x_{3}=0$.) This gives many pairs of disjoint properly immersed
minimal surfaces in $\mathbb{R}^{4}$ of any given conformal type. There also
exist pairs of disjoint catenoids in $\mathbb{R}^{4}$ whose ends are
asymptotic to a pair of orthogonal $2$-planes in $\mathbb{R}^{4}$, so their
closures in $\mathbb{RP}^{4}$ are disjoint as well. On the other hand, a pair
of complex algebraic curves in $\mathbb{C}^{2}$ intersect, or their closures
in $\mathbb{CP}^{2}$ intersect at infinity. (Note that every complex curve is
also a minimal surface.)

### Acknowledgements

I thank Antonio Alarcón and Joaquín Pérez for information on properly embedded
minimal surfaces with bounded curvature in $\mathbb{R}^{3}$, and for general
advice. I also thank G. Pacelli Bessa and Leandro F. Pessoa for communication
regarding their paper [9] and the paper by Gama et al. [17]. Finally, my
sincere thanks go to an anonymous referee for the comments and suggestions
which led to an improved presentation.

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