# A Review of Scene Representations for Robot Manipulators

Carter Sifferman<EMAIL_ADDRESS>
Department of Computer Sciences
University of Wisconsin - Madison

##  1 Introduction

For a robot to act intelligently, it needs to sense the world around it.
Increasingly, robots build an internal representation of the world from sensor
readings. This representation can then be used to inform downstream tasks,
such as manipulation, collision avoidance, or human interaction. In practice,
scene representations vary widely depending on the type of robot, the sensing
modality, and the task that the robot is designed to do. This review provides
an overview of the scene representations used for robot manipulators (robot
arms). We focus primarily on representations which are built from real world
sensing and are used to inform some downstream robotics task.

Building an intermediate scene representation is not necessary for a robotics
system. It is completely possible for a robotics system to act directly on
sensor data (e.g. predict appropriate grasps directly from RGB images), and we
will look at many such systems within this review. However, we argue that
intermediate scene representations are beneficial for robot manipulators as
they:

  * •

act as spatial memory

  * •

are efficient storage of past memories

  * •

allow long-horizon planning

  * •

can act as regularization and encode spatial priors for learning systems

Figure 1: This review focuses on robotics applications which build some
intermediate scene representation between sensor data and a downstream
robotics task. Image sources: [1, 2, 3]

In this review, we organize scene representations into three categories
depending on the task that the representation supports. These categories make
up the sections of our review: collision avoidance (section 2), manipulation
(section 3), and teleoperation (section 4). Within each section, we provide a
review of the existing literature, summarize the challenges in the area, and
consider directions for future research. In section 5, we look at scene
representations for manipulators as a whole, and consider directions for
future research which cut across our three categories.

###  1.1 Literature Survey Process

Our literature survey consisted of two phases. In phase 1, we performed a
broad search of existing reviews in order to gain context and understand the
broader landscape of robotics. To find these reviews, we use Google Scholar
search with the “Review Articles” filter enabled. The result of this search is
13 reviews spanning a broad spectrum: manipulation and grasping [4, 5, 6, 7,
8, 9, 10], SLAM [11], human-robot interaction [12, 13], teleoperation [14],
motion planning [15, 16] and inverse kinematics [17]. Knowledge gained from
these reviews was used to determine our taxonomy of scene representations, and
the scope of this review.

The goal of phase 2 of our survey process was to find directly relevant papers
which employ a scene representation for robot manipulators. As the papers that
we search span a wide range of topics, keyword searches did not prove
effective. Instead, we “snowballed” through references, beginning at seed
papers which were found through keyword search, reviews, or consulting with
colleagues. These seeds are shown in table 1. We snowballed through references
both through traditional reference chasing, and using Google Scholar’s “Cited
By” page to find papers published after the seed paper. We also use the
Abstract Viewer Project111https://pages.graphics.cs.wisc.edu/AbstractsViewer/,
a system for finding related papers developed as a research project at UW-
Madison. Abstract Viewer does not use citations to find related papers,
instead relying on analysis of textual content. This proved helpful for
finding still-related papers when references ran dry. In total, 69 papers were
collected in phase 2.

This review is not meant to act as a comprehensive survey of any one subject.
Instead, we hope to give a broad overview of the field and provide jumping-off
points for further reading. Articles were not selected to be a representative
sample of the entire field; papers which use an identical scene representation
to existing work may be excluded while those with a unique scene
representation are generally included.

Category | Snowball Seeds | # Papers Found
---|---|---
Collision avoidance | [2, 18, 3] | 26
Manipulation | [19, 20, 21] | 28
Teleoperation | [22, 23, 14] | 16
Table 1: The three sections that this review is organized into, and the
“snowball seeds” which began the literature review process.

##  2 Collision Avoidance

A central challenge in robotics is being able to avoid collisions, which can
potentially be very costly with a powerful, fast, and expensive robot. Some
approaches for collision avoidance respond directly to measurements from
robot-mounted sensors, without building any intermediate representation [24,
25, 26]. A downside of this approach is that it has no memory: the robot can
only act based on what it currently observes, and cannot plan based on its
previous observations. As a result, these approaches are overly cautious and
perform poorly in challenging conditions.

Modern approaches for collision avoidance can be broadly grouped into two
categories: motion planning [27], in which the start and goal position of the
robot end effector are known, and the goal is to find a viable, collision-free
path between the two, and inverse kinematics, in which the end effector
position is known, and the goal is to find a viable joint configuration of the
robot which matches that end effector position. This distinction is
unimportant for this review, as both of these approaches have the same
requirements of their scene representations: fast collision checks and
(sometimes) fast calculation of the distance to the nearest obstacle or
direction to the nearest obstacle. As a result, scene representations for
collision avoidance are similar whether the problem is formulated as inverse
kinematics or motion planning.

Potential Fields– Early approaches to collision avoidance in a motion planning
context represented the environment with a potential field, first proposed in
[28]. This potential field can be evaluated at any point to yield a scalar
“potential” value. This value is determined by both the scene geometry (which
have high potential around them) and the desired location (which has a low
potential). In order to move through space, the robot simply follows the
negative gradient of this potential field. The potential field was heavily
utilized in early motion planning work [29, 30, 31]. While effective for the
time, potential fields suffer from a few problems: the potential function is
prone to local minima, and can be very difficult and computationally intensive
to construct [15]. Additionally, it is impractical to construct potential
fields in real-time from sensor measurements, both because they are slow to
construct, and because they require scene geometry to be described in a
friendly closed-form, which sensors cannot provide natively. Later work [32]
improved on the local minimum problem by taking into account the relative
starting position as well as scene geometry during potential field
construction, but nonetheless potential fields have fallen out of favor in
collision avoidance applications since the early 2000s.

Signed Distance Fields– A signed distance field is a mapping between 3D points
in space x and the scalar distance $d$ to the nearest obstacle:

$SDF(\textbf{x})=d$

The SDF has the nice property that taking $-\nabla SDF(\textbf{x})$ yields a
vector pointing towards the nearest obstacle. This representation has been
used in computer graphics since at least 1998 [33, 34], and became popular for
robot collision avoidance with the introduction of the highly influential
CHOMP motion planner in 2009 [3]. CHOMP uses the signed distance field, along
with pre-computed gradients to perform optimization over the robot
configuration. Subsequently, the popular STOMP [35], TrajOpt [36], and ITOMP
[37] motion planners also used a signed distance field to represent their
environment, and used gradients in a similar way. In each of these works, the
signed distance field is computed at fixed points on a regular voxel grid as a
pre-processing step. To do such a computation, a precise model of the
underlying geometry is needed, typically in the form of a mesh. Similarly to
potential fields, computing a signed distance field is computationally
expensive, and generating it from noisy sensor data is difficult. In practice,
collision avoidance approaches which use an SDF are constrained to
simulations, where the SDF can be pre-computed, or static environments in the
real world. Regardless, SDFs are the most popular scene representation for
collision avoidance, largely because they are supported by popular and
effective motion planners. There exist libraries such as VoxBlox [38] and
FIESTA [39] which efficiently compute and store discretely sampled SDFs for
this purpose.

Collections of Primitives– A less common method for representing geometry for
collision avoidance is with a collection of primitive shapes, such as spheres,
cylinders, and cubes. These primitives are usually stored parametrically, so
that collision checking can be done quickly and the objects represented
natively in optimization solvers. Toussaint et. al. [40] proposed Logic-
Geometric Programming, in which the scene is composed entirely of parametric
cylinders, blocks, and planes. This paper has been influential for its elegant
optimization-centered formulation, but the scene representation used within
has not been heavily utilized; it serves more as a demonstration of the
approach’s capability. Similarly, Gaertner et. al. [41] considers collision
avoidance with humanoid robots, and uses a collection of primitives to
represent dynamic scenes. Similarly to Toussaint, the collection of primitives
is used primarily to demonstrate the capabilities of the system. In contrast,
Zimmerman et. al. [42] proposes a method for using collections of primitives
in gradient-based optimization methods such as TrajOpt [36]. They provide a
unified method for dealing with many types of primitives, and a method for
taking the derivative of the distance to the nearest primitive, making a
collection of primitives a drop-in replacement for SDFs. However, this
approach has not seen widespread adoption.

Collections of Convex Hulls– A collection of convex hulls is another less
common way to represent scene geometry for collision avoidance. Similarly to
primitive shapes, convex hulls allow for fast collision checking and natural
representation in optimization solvers. Convex hulls have the additional
benefit that any shape can be broken down to a collection of convex hulls via
an algorithm like QuickHull [43]. Schulman et. al. [44] uses a set of convex
hulls to represent a scene, and outperforms the SDF-based motion planners of
the time like CHOMP [3] and STOMP [35]. CollisionIK [18] introduces an
optimization-based method for inverse kinematics, which is able to operate in
real-time (e.g. for mimicry control) and avoid collisions with dynamic
obstacles. CollisionIK mentions that point cloud objects could be broken down
into convex hulls in real time, but does not demonstrate such a process. To
our knowledge, no existing approach constructs convex hulls in real time from
sensor data.

Learned Representations– Within the last year, one approach has emerged which
uses a learned environment representation to enable real-time collision
avoidance with dynamic obstacles and real-world sensing. This paper is
somewhat influenced by the growing literature around learned representations
in computer vision [45], graphics [46], and SLAM [47, 48]. RCIK [2] proposes a
collision cost prediction network, a neural network which takes as input
features extracted from an occupancy grid, as well as a 3D point in that grid;
from this input the network predicts the collision cost, which is an
approximation of the SDF evaluated at that 3D point. The occupancy grid can be
generated in real time with one or more depth cameras. The network is trained
on one million simulated examples of random environments and joint
configurations. While this method does not have the collision avoidance
guarantees provided in theory by other methods, it is the first method to
perform collision-free inverse kinematics in real time with real sensing. This
same approach was later evaluated under real-time control [49].

Figure 2: The collision cost prediction network of RCIK [2] is trained on
simulated data, and outputs a collsion cost $F_{col}$ which approximates the
signed distance function to enable real-world real-time collision avoidance.

###  2.1 Future Directions in Collision Avoidance

Real-time generation of signed distance fields. Signed distance fields have
proven highly effective for enabling collision avoidance. However, SDFs are
very costly to generate, and require a very accurate representation of the
underlying environment. Because of this, the vast majority of work on
collision avoidance with robot manipulators does so only in simulation, or in
manually recreated static environments. A pressing challenge is finding ways
to bring these methods to the real world by constructing an SDF in real time.
Recent work on neural representations has enabled real-time construction of a
neural SDF from depth imagery called iSDF [50]. Similarly to RCIK [2], the SDF
produced by iSDF is only an approximation, however it is generated over time
from multiple sensor observations, and does not rely on simulated pre-
training. Adapting a similar approach for static or manipulator-mounted depth
cameras could enable real-world operation of the many collision avoidance
approaches which rely on SDFs.

Moving beyond signed distance fields. Signed distance fields alone are great
for collision avoidance, but offer some limitations. For example, not all
collisions are equally costly. Colliding with a pillow might be admissible if
it means avoiding collision with a human. Future representations, and
algorithms which act on them, could store semantic information along with
scene geometry, to enable such decisions to be made. Learning such semantic
scene properties may be possible via extensive pre-training, or via
interaction.

##  3 Manipulation

Arguably the most important task for a robot manipulator is to manipulate
things. Manipulation can mean grasping with a simple one degree-of-freedom
gripper, articulated grasping, or simple pushing and nudging of objects with
any part of the robot. Robot manipulation is a vast field, with approaches
specialized for dealing with many specific challenges. To keep the scope of
this review reasonable, we focus on scene representations which are used for:

  * •

Basic grasping with a 1DoF gripper

  * •

Generalizable grasping

  * •

Articulated grasping

  * •

Predicting scene flow

Direct Action on Images– While this review focuses on intermediate scene
representations, it is clear that, for the purposes of robot manipulation, an
intermediate representation is not necessary. A seminal work in this area was
Saxena et. al. [21] in 2006, which was the first work to directly predict
grasping points from an image. They use a neural network to, given an RGB
image of an object to be grasped, predict pixels in the image at which the
object is most suitable for grasping. To train their neural network, they use
supervised learning on a large fully synthetic dataset. Two years later, the
same authors improved on the approach by using RGB-D imagery as input to the
network [51]. Lenz et. al. [52] improved upon previous work by aiming to
predict the single best grasp, rather than listing many viable ones, and
predicting the orientation and extent of the grasp along with the location.
Later, the “DexNet” series of papers offered iterative improvements by
improving training data and tweaking the neural network outputs, as well as
considering alternative grip types such as articulated hands and suction cups
[53, 54, 55]. Other work aims to grasp objects given some semantic label, e.g.
“coffee mug” or “plate”, this is sometimes called semantic grasping. Jang et.
al. [56] proposes a two-stream approach to semantic grasping, in which one
stream identifies objects while another finds suitable grasps. Schwarz et. al.
[57] demonstrates a semantic grasping pipeline which uses a suction cup
gripper and works by simply segmenting out objects and finding their center of
mass. This approach performed well at the highly competitive Amazon Picking
Challenge222https://robohub.org/amazon-picking-challenge/. These direct
prediction approaches are highly effective for real world operation, however
they are fairly limited in their potential. These approaches can only perform
single-shot grasping, meaning they are unable to, for example, rotate an
object, let go of it, and grab it again. They also have a limited ability to
reason about novel shapes in 3D. Lastly, these approaches typically rely on
some synthetic training data, which must be generated via other 3D-aware
methods.

Meshes– A number of works use a 3D triangular mesh to represent objects for
manipulation. The problem of finding suitable grasps given a 3D mesh is a
long-standing problem with active research [58, 59, 60, 61, 62]. This
paragraph focuses not on those algorithms, but on real-world systems for
manipulation which represent objects as meshes. The first of such real-world
systems was Berenson et. al. [63] in 2007. This work makes grasping possible
in the real world by incorporating information about not only the object to be
grasped, but also nearby obstacles, such as the table or other objects, into
the grasp selection algorithm. In order to sense the positions of objects in
real-world tests, this approach relies on motion capture markers being placed
on each object, as shown in fig. 3. Goldfeder et. al. [64] introduced a method
for finding good grasps given a mesh, and tested their method by scanning real
objects. This approach was somewhat effective, but the conversion from scan to
mesh does not happen in real time. Collet et. al. [65] circumvents the problem
of real-time mesh construction by modeling each object as a primitive, and
fitting that primitive to a point cloud in real time. Later work by the same
authors [66] extends this idea to arbitrary meshes by using a 6D pose
recognition algorithm. Assuming that a mesh of the object is known, this
approach enables prediction of the mesh’s pose in real time. Papazov et. al.
[67] takes a similar approach, but assumes that the object is represented with
a set of points and surface normals, rather than an RGB image. Varley et. al.
[68] removes the requirement of a pre-made mesh by teaching a neural network
to complete a point cloud. From the completed point cloud, a mesh can be built
and that mesh passed off to a mesh-based grasp planner in real time. The steps
in Varley et. al.’s pipeline are shown in fig. 4. In each of these approaches,
we see that the limiting factor is not the grasp planning algorithms
themselves, but the generation of meshes in real time.

Figure 3: An early approach for real world grasping, Berenson et. al. [63],
relied on a motion capture system and pre-defined meshes to perform grasping
in the real world. Figure 4: Varley et. al. [68] uses a neural network to
enable shape completion on partial point cloud observations. The completed
shapes can then be transformed to a mesh and used in any mesh-based grasp
planner. Image from Varley et. al.

Point Clouds– Point clouds have been used as the basis for grasp planning in a
similar manner to meshes. In constrast to meshes, point clouds are closer to
the native output of commonly used depth cameras, making them more practical
for real world construction. Approaches exist for grasp planning directly on
point clouds [69, 70, 71, 72], although less numerous than those for meshes.
Florence et. al. [73] (covered in more depth in the following paragraph)
introduces a method for finding corresponding grasps between similar objects,
and uses point clouds to perform the grasping in real-world examples. Simeonov
et. al. [74] considers the problem of manipulation planning, and is able to
predict the movement of scene objects directly from their point clouds. This
prediction is used to plan for manipulations which move the scene towards a
goal state.

Voxel Grids– Zhang et. al. [75] constructs a gripper-sized voxel grid from a
point cloud and a given gripper position, with each voxel encoding whether the
space is occupied by a point in the point cloud. This voxel grid can then be
compared (via nearest neighbors) to previously simulated voxel grids to
determine whether it corresponds to a good gripper position.

Learned Representations– Dense object nets [73] are an approach for
generalizable grasping. For a given input image, a neural network is trained
to produce a dense pixel-level feature map which produces similar output
feature vectors for semantically similar points in multiple images of similar
objects. For example, for any two images of a mug, the goal is that pixels
corresponding to the same point on the mug handle will have the same feature
vector in the output representation. This representation is trained to be
consistent across object instance, pose, and deformation, enabling
generalizable grasping. They demonstrate that this approach is effective at
generalizable grasping in the real world, using merged point cloud data from
multiple depth cameras. Simeonov et. al. [19] expands upon this approach by
taking a 3D point as input to the neural network, rather than a 2D pixel
position. The object properties are encoded within the weights of a multi-
layer perceptron, similarly to NeRF [45]. They demonstrate that this approach
can be used to perform pick-and-place tasks of novel objects of a given class
given fewer than 10 demonstrations. Additionally, the architecture of their
MLP ensures that the descriptor fields are SE(3)-equivariant, making them
robust to arbitrary object poses. Van Der Merwe et. al. [76] uses a learned
representation for articulated grasping. A neural network is trained to, given
a point cloud and a 3D query point, approximate the signed distance function
at that point. The latent space of this network is concatenated with
information about desired grasp qualities and robot configuration, and fed
into another network which predicts the success rate of the proposed grasp. Xu
et. al. [20] builds a visual predictive model for robotics. Their goal is to,
given some representation of the scene along with a robot action, predict the
scene after the action has occurred. Rather than using a manually constructed
scene representation, they allow the scene representation to be learned in an
end-to-end manner, as a 128x128x48 feature grid with an 8-dimensional vector
at each point in the grid. Using this representation, they are able to predict
3D scene flow and plan manipulation tasks.

###  3.1 Future Directions in Manipulation

Performing 3D grasp planning based on real-world sensor data. There is a
disconnect between the way scenes are represented for grasp planning
(primarily meshes) and the way that the most effective robotics systems, such
as those used in the Amazon Picking Challenge, are operated (primarily direct
prediction). An important direction for future work is bridging this gap.
Direct prediction methods are severely limited in their spatial reasoning,
while mesh-based methods are severely limited in their real-world operation.
Point cloud manipulation and learned representations may bridge the gap, but
do not currently offer the high grasping accuracy of mesh based approaches.

Representing complex object properties. Any object may have many properties
which are relevant to manipulation: its center of mass, deformation
properties, or constraints on its motion (e.g. hold a mug full of coffee
upright, pull a drawer straight out). Learning-based approaches, such as Dense
PhysNet [77] have shown promise towards being able to learn these properties
autonomously. A needed direction for future research is determining ways to
learn these properties, generalize them to novel objects, and store these
properties alongside their geometric representations.

##  4 Teleoperation

In a teleoperation scenario, a human controls a robot remotely, and relies on
an intermediate interface, such as a monitor or VR headset, to understand the
robot’s surroundings. Much recent work on teleoperation places the human
operator in virtual reality (VR); studies have shown increased task
performance in virtual reality, due to the ease of controlling the user’s view
and all six degrees-of-freedom of the robot [78]. However, the scene
representations used in these VR approaches can generally applied to any sort
of teleoperation scenario.

###  4.1 Human Control of Mobile Robots

Mobile robots are much more likely than robot manipulators to venture into
unknown environments. Because of this, the foundational work in representing
scenes to remote operators has taken place in mobile robotics. For context, we
offer a brief overview of such work here. Nielsen et. al. [79] was an early
attempt at incorporating data beyond RGB camera feeds into robot
teleoperation. They create a dashboard which shows an RGB camera feed along
with LiDaR scans and a rough map of the environment. While all of the
information is presented to the user, it is not combined to create an
intuitive display. Kelly et. al. [80] builds a 3D map of the environment by
using LiDaR scans to create a 3D map, and coloring that map with data from RGB
images. A CAD model of the mobile robot is placed in the environment. This
allows rendering arbitrary viewpoints, such as an overview of the entire
scene, an overhead view, or a third person view, as would be seen in a racing
video game. Stotko et. al. [81] builds a 3D mesh-based scene representation
for a mobile manipulator, and displays the mesh to the user interactively in
virtual reality. They find that users have fewer collisions, and report a
greater level of immersion and awareness than with a 2D interface. Livantino
et. al. [23] augments robot-attached camera views with 3D data to overlay data
such as desired path, destination, and label traversable terrain. For mobile
robot teleoperation, the trend has been towards a free floating, user
controllable view and natural display of information. These same goals apply
to robot manipulator teleoperation.

(a) Dashboard-based interface from [82] (2004)

(b) Augmented interface from [79] (2007)

(c) 3D interface from [80] (2011)

Figure 5: A history of the typical information displays used in mobile robot
teleoperation

###  4.2 Human Control of Robot Manipulators

No intermediate representation– A simple baseline for teleoperation is the use
of one or more static cameras, which the user can either see all at once, or
switch between manually. While simple to implement, a static camera approach
is limiting: they present the user with limited geometric information, and
don’t leverage computation to enhance human perception. Static cameras are
also prone to being blocked by the robot manipulator itself. There are a
number of approaches which improve on the static camera baseline without
building an intermediate representation. Murata et. al. [83] considers mobile
manipulators (manipulators mounted on mobile robots), and renders a CAD model
of the robot on top of a background made of stitched, observed images. The CAD
model is kept updated to represent the robot’s current state, and the user is
able to position the camera to generate an arbitrary view. A different
approach is taken by Rakita et. al. [84], in which a robot arm is used as a
camera operator for another robot arm. The camera operator’s pose is
automatically optimized in real time to present a clear view of the
manipulating robot’s end effector.

Point Clouds– Point clouds are a common choice for representing scenes for
real time operation, because they are the native output of depth cameras, and
can be displayed in real time with little processing. Brizzi et. al. [85]
considers augmented reality for VR teleoperation. Operators see an RGB image
of the scene, along with features, such as distance to the target, direction
to the target, or gripper state. Point clouds from a depth camera are used to
calculate these features. Kohn et. al. [86] displays a combination of point
clouds from RGBD cameras and meshes to a user in VR. Meshes are used for known
objects (such as the floor, table, and robot) and point clouds are used for
the unknown. The unknown objects are filtered in real time according to the
meshes, as shown in fig. 6. A similar approach is taken by Su et. al. [87].
Wei et. al. [22] similarly displays point clouds to users, but unlike previous
approaches, they do not use meshes to represent known objects, aside from the
robot gripper. They perform a user study, comparing the point cloud to multi-
view RGB and to a point cloud projected onto an RGB image. In their
experiments, the hybrid point cloud and RGB view performs best.

Figure 6: Kohn et. al. [86] uses meshes to represent known objects (table and
robot) and point clouds to represent unknown objects (puzzle box) for
teleoperation. Image from Kohn et. al.

Meshes– While point clouds are native and fast to display, meshes may be
preferred due to their potentially higher visual fidelity. Wonsick et. al.
[88] takes an approach similar to the mesh-based approaches in section 3; a
point cloud of the scene is semantically segmented to identify its class
amongst known objects. A known 3D model is then fit to each point cloud
segment, and that 3D model is displayed to the user in virtual reality. Models
are able to be constructed in much higher fidelity, and the scene can be
entirely rendered, enabling control of lighting, texture, etc. They run a user
study and say that users find their approach more usable and less cognitive
load than point-cloud streaming alone. However their approach relies on having
known 3D models of objects in the environment, and has no fallback if such a
model does not exist, as such it is not suitable to unknown environments.
Piyavichayanon et. al. [89] generates a “depth mesh” by combining the view of
multiple depth cameras. This mesh is used to display augmented reality
features, such as distance to a collision state, on a handheld smartphone.
Merwe et. al. [90] performs a user study to investigate how the type of
information presented to the operator changes their performance. They compare
a full information 3D model to a “representative” mesh based model, which only
displays crucial information. They find that task completion time is lower
with the full model, but cognitive load is higher.

Occupancy Grids– Omarali et. al. [91] considers multiple modes of
visualization for VR teleoperation. Alongside the depth camera-based
baselines, they present a hybrid view, which displays the current output from
the depth camera, alongside a translucent occupancy map in the previously
observed areas. This occupancy map gives the user some context for the greater
environment, while indicating that the context is not as reliable as the
currently observed region. They find that users prefer the hybrid depth camera
+ occupancy map approach.

###  4.3 Future Directions in Teleoperation

Improving real-time visualization of the robot’s environment. Existing
approaches tend to rely on depth cameras, which can render the world in real
time, but are low in detail and high in noise. Approaches which attempt to
process this data into something more appealing, like a mesh, require heavy
computation and require meshes of potential objects to be known a priori.
Future work should consider ways to improve the fidelity and detail of
reconstruction in real time and in unknown environments. To some extent,
advances in imaging, such as high resolution depth sensors may improve this
problem. Another potential solution is dynamically directing sensing towards
areas that need high detail, such as around the end effector. The problem of
representing the environment from a novel perspective is known as novel view
synthesis and is well studied in the computer vision literature (see the
recent review [92]). Incorporating techniques from novel view synthesis could
allow a more complete environment representation to be displayed to the
operator.

Building representations suited to the operator. Currently, the vast majority
of approaches focus on building representations which represent the
environment faithfully. However, it has long been known that realistic
displays are not always ideal, as they can make it difficult to parse what is
relevant [93]. Some work covered in this review has considered building a
sparse representation of only relevant features [90, 88]. However, these
approaches require such a representation to be specified by hand prior to
operation. Future work should consider ways of filtering the scene
representation to represent only what is important to the user in an unknown
scene.

##  5 Future Directions

Quantifying Uncertainty. Present systems for robot manipulation build a most
likely map of the environment. In poor sensing conditions, this map may be
highly unreliable, leading the downstream robotics application to make
incorrect but fully confident choices based on the unreliable map. Future work
should work to incorporate uncertainty into the scene representation itself,
in order to enable robot policies which act intelligently in the face of
uncertainty. When in a highly uncertain environment such a robot could, for
example, ask a human for help, or gather additional information about its
environment before taking action. The issue of uncertainty estimation is well
studied in mobile robotics, especially in the context of SLAM [94]. Recent
approaches in novel view synthesis have also enabled dense estimates for
geometric and photometric uncertainty [95, 96], as shown in fig. 7. Such
approaches could be applied to robot manipulators, especially for problems
like collision avoidance.

Figure 7: Shen et. al. [95] builds a scene representation which includes dense
uncertainty estimates. A similar approach could prove useful to build scene
representations for robot manipulators. Image from Shen et. al.

Joint prediction of scene semantics and geometry. Present approaches which
infer scene semantics do so by taking some geometric representation of the
scene as input, and producing a semantic map based on that geometry. With this
approach, scene geometry informs scene semantics, but semantics has no
influence on geometry. In reality, semantic information about an object can
provide queues about its likely geometry. If an object is identified as a
coffee mug, it probably has a handle on it. If an object is identified as a
baseball, it is probably spherical. Floors and tabletops tend to be flat and
level with one another. Future approaches could infer scene geometry and
semantics jointly, by encoding sensor data in a learned latent representation,
which is used to inform both geometric and semantic inference. Joint inference
approaches have been used for semantic SLAM [97, 98, 99]. Implementing such an
approach for robot manipulators would enable higher fidelity predictive
mapping and improved semantic understanding.

Sensing-first development and real-world benchmarking. Many existing
approaches, particularly for grasping and collision avoidance, are benchmarked
entirely in simulation. Real-world demonstrations are addressed after the
development has been optimized for simulation, and often only occur under
highly controlled conditions. Future work should consider developing methods
with sensing in mind from the beginning, and aiming for high performance on
real-world tasks with sensing, rather than in simulation. This means building
systems which are robust to sensor noise, and able to act on raw sensor data
in real time. One encouraging sign towards such a goal is the just announced
RT-1 project from Google [100], which is an embodied agent that is benchmarked
entirely on real-world tasks.

Representations for Alternative Sensing Modalities. All works addressed in
this survey utilize data from typical vision based sensors: RGB cameras, depth
cameras, and/or LiDaR. There are many other sensor types used in robotics,
such as force torque sensors and tactile sensors, which provide information
about the environment. However, existing approaches react immediately to
information from these sensors, and do not build an intermediate
representation which persists over time. Can we build scene representations
which model the factors that these sensors measure? For example, the
measurement from a tactile sensor may be influenced by how hard the object is,
how heavy it is, or even how conductive it is, alongside the object’s
geometry. Present scene representations do not encode all of this information,
making it very difficult to relate new observations from a tactile sensor to
the known scene. Creating scene representations which do encode information
from alternative sensors could enable applications such as SLAM or collision
avoidance under low vision conditions.

##  6 Conclusion

Building new scene representations for robot manipulation is an important step
towards creating fully autonomous embodied agents which can interact
intelligently with the world. Continued advances in sensing and robotics will
necessitate new representations which encode data from new sensors and support
new robot form factors and applications. There are two challenges which are
present in all works covered in this review: building representations which
can be constructed in real time, and which are robust to sensor noise. Given
the recent pace of advancement in robotics and computing, we are confident
that these challenges will be sufficiently overcome.

Application |  Common
Representations |  Less Common
Representations
---|---|---
Collision Avoidance |  Signed distance fields |  Convex hulls
Potential fields
Geometric primitives
Learned representations
Manipulation and Grasping |  Direct action
Meshes
Point clouds |  Voxel grids
Signed Distance Fields
Learned representations
Teleoperation |  Point clouds |  Meshes
Occupancy grids
Table 2: Scene representations covered in this review

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111Lina Wu is partially supported by National Natural Science Foundation of
China (12201030), China Postdoctoral Science Foundation (2022M720394) and
Talent Fund of Beijing Jiaotong University (2022RC028).

# Simple blow-up solutions of singular Liouville equations

Lina Wu  Lina Wu
School of Mathematics and Statistics
Beijing Jiaotong University
Beijing, 100044, China<EMAIL_ADDRESS>
###### Abstract.

In a recent series of important works [25, 26, 27], Wei-Zhang proved several
vanishing theorems for non-simple blow-up solutions of singular Liouville
equations. It is well known that a non-simple blow-up situation happens when
the spherical Harnack inequality is violated near a quantized singular source.
In this article, we further strengthen the conclusions of Wei-Zhang by proving
that if the spherical Harnack inequality does hold, there exist blow-up
solutions with non-vanishing coefficient functions.

###### Key words and phrases:

Liouville equation, quantized singular source, non-simple blow-up,
construction of solutions, blow-up solutions. Spherical Harnack inequality

##  1\. Introduction

It is well known that the following Liouville equation has a rich background
in geometry and Physics.

(1.1) $\Delta
u+h(x)e^{u(x)}=\sum_{t=1}^{L}4\pi\gamma_{t}\delta_{p_{t}}\quad\mbox{in}\quad\Omega\subset\mathbb{R}^{2},$

where $\Omega$ is a subset of $\mathbb{R}^{2}$, $p_{1},..p_{L}$ are $L$ points
in $\Omega$ and $4\pi\gamma_{t}\delta_{p_{t}}$ ($t=1,...,L$) are Dirac masses
placed at $p_{t}$. Since applications require integrability of $e^{u}$ we
assume $\gamma_{t}>-1$ for each $t$.

Equation (1.1) is one of the most extensively studied elliptic partial
differential equations in recent years. In conformal geometry, (1.1) is
related to the well-known Nirenberg problem when all $\gamma_{t}=0$. The
recent progress on this project can be seen in Kazdan-Warner [15], Chang-
Gursky-Yang [5], Chang-Yang [6], Cheng-Lin [10], and the references therein.
If some $\gamma_{t}\neq 0$, (1.1) arises from the existence of conformal
metric with conic singularities, seen in Fang-Lai [13], Troyanov [22], Wei-
Zhang [23]. Also, it serves as a model equation in the Chern-Simons-Higgs
theory and in the Liouville system, the interested readers may browse
Chanillo-Kiessling [7], Spruck-Yang [19], Tarantello [20], Yang [28], and the
references therein.

It is well known that if there is no singularity in (1.1), $h\equiv 1$ and
$\int_{\mathbb{R}^{2}}e^{u}{\rm d}x<\infty$, a global solution belongs to a
family described by three parameters (see [9]). Then Y. Y. Li [17] proved the
first uniform approximation theorem, which confirms that around a regular
blow-up point, the profile of a blow-up sequence is close to that of a
sequence of global solutions. Later Chen-Lin [8], Zhang [29], Gluck [14],
Bartolucci, et,al [2] improved Li’s estimate by obtaining better pointwise
estimates and some gradient estimates. It turns out that the blow-up point has
to be a critical point of a function determined by the coefficient function.
This plays a crucial role in applications. In the non-quantized case, the
classification theorem was proved by Prajapat-Tarantello, the uniform estimate
is obtained by Bartolucci-Chen-Lin-Tarantello [1], Bartolucci-Tarantello [3],
Zhang [30]. The most difficult case is when the singular source is quantized.
In this case, the first breakthrough was obtained by Kuo-Lin in [16], then
independently by Bartolucci-Tarantello in [3]. In this case, if the spherical
Harnack inequality is violated near a quantized singular source, the profile
of bubbling solutions appears to have multiple local maximums. Here a sequence
of bubbling solutions satisfying _spherical Harnack inequality_ means the
oscillation of solutions on each fixed radius around the singular point is
uniformly bounded. In the work of Kuo-Lin, they use _non-simple blow-up_ to
describe this phenomenon. In a recent series of works of Wei-Zhang [25, 26,
27], they proved the first vanishing theorems for the non-simple blow-up case.
Their two main results can be stated as follows:

Let $\\{u_{k}\\}_{k=1}^{\infty}$ be a sequence of blow-up solutions of

(1.2) $\Delta u_{k}+|x|^{2\alpha}h_{k}(x)e^{u_{k}(x)}=0,\quad\mbox{in}\quad
B_{1}$

where $h_{k}$ is a sequence of smooth, positive functions in $B_{1}$:

(1.3) $\frac{1}{c_{1}}\leq h_{k}(x)\leq
c_{1},\quad\|\nabla^{\beta}h_{k}(x)\|_{B_{1}}\leq c_{1},\quad x\in
B_{1},\quad|\beta|=1,2,3.$

for some $c_{1}>0$. Let $0$ be the only blow-up point of $u_{k}$ in $B_{1}$,
and suppose $u_{k}$ has a bounded oscillation on $\partial B_{1}$:

(1.4) $|u_{k}(x)-u_{k}(y)|\leq C,\quad\forall x,y\in\partial B_{1},$

and a uniform bound on its integration:

(1.5) $\int_{B_{1}}|x|^{2\alpha}h_{k}(x)e^{u_{k}(x)}{\rm d}x<C$

for some $C>0$ independent of $k$. In their first vanishing theorem Wei-Zhang
proved that

_Theorem A: (Wei-Zhang). Let $u_{k}$ be a sequence of non-simple blow-up
solutions around the origin. Suppose $0$ is the only blow-up point in $B_{1}$
and $u_{k}$ satisfies (1.2),(1.4) and (1.5). Then along a sub-sequence_

$\lim_{k\to\infty}\nabla(\log h_{k}+\psi_{k})(0)=0$

_where $\psi_{k}$ is the harmonic function that eliminates the finite
oscillation of $u_{k}$ on $\partial B_{1}$:_

(1.6) $\Delta\psi_{k}=0,\quad\mbox{in}\quad
B_{1},\quad\psi_{k}(x)=u_{k}(x)-\frac{1}{2\pi}\int_{\partial B_{1}}u_{k},\quad
x\in\partial B_{1}.$

In their recent work, Wei-Zhang further proved the following Laplacian
vanishing theorem:

_Theorem B: (Wei-Zhang). Let $u_{k}$ be the same as in Theorem A. Then along a
subsequence,_

$\lim_{k\to\infty}\Delta(\log h_{k})(0)=0.$

It is important to point out that in both Theorem A and Theorem B, the blow-up
sequence has to be _non-simple_ , this assumption implies that
$\alpha\in\mathbb{N}$. Both Theorem A and Theorem B are powerful tools in
application, since the equation (1.2) represents a number of situations in
more general equations/systems. For example, in the author’s recent joint work
with Wei and Zhang [24], we proved that under certain conditions on the
coefficient function and Gauss curvature, all blow-up points to Toda systems
are simple.

The purpose of this article is twofold. First if $\alpha=0$ in (1.2) and $0$
is the only blow-up point, it is well known that (see [8, 14, 29]) along a
sub-sequence $\lim_{k\to\infty}\nabla(\log h_{k}+\psi_{k})(0)=0$. Over the
years it has long been suspected that this property does not hold if $\alpha$
is not an integer. This is indeed verified in our first main theorem:

######  Theorem 1.1.

For any given $\alpha>-1$ and $\alpha\not\in\mathbb{N}\cup\\{0\\}$, there
exist a sequence $h_{k}$ satisfying (1.3) and

(1.7) $|\nabla\log h_{k}(0)+\nabla\psi_{k}(0)|\geq c_{1},\quad|\Delta\log
h_{k}(0)|\geq c_{1}$

for some $c_{1}>0$, Corresponding to $h_{k}$ there is a sequence of blow-up
solutions $u_{k}$ of (1.2) such that the origin is its only blow-up point,
(1.4) (1.5) holds for $u_{k}$, which also satisfies the spherical Harnack
inequality around the origin.

The second goal is to prove that when $\alpha\in\mathbb{N}\cup\\{0\\}$ we can
construct a sequence of _simple_ blow-up solutions that does not satisfy the
Laplacian vanishing theorem.

######  Theorem 1.2.

Let $\alpha\in\mathbb{N}\cup\\{0\\}$, there exist a sequence of blow-up
solutions $\\{u_{k}\\}_{k=1}^{\infty}$ of (1.2) having $0$ as its only blow-up
point in $B_{1}$. Moreover $\\{u_{k}\\}$ satisfies (1.4) (1.5) and the
coefficient $h_{k}$ satisfies (1.3) and

$|\Delta(\log h_{k})(0)|\geq c,\,\,\mbox{for a constant
}\,\,c>0\,\,\mbox{independent of $k$}.$

Theorem 1.1 settles the conjecture that around a non-quantized singular
source, the vanishing theorems do not hold. Theorem 1.2 proves that it is
essential to have a _non-simple_ blow-up sequence in Theorem B. If this
assumption is violated, the corresponding Laplacian vanishing property also
fails. However, this article did not provide a similar example for the first-
order vanishing theorem in Theorem A.

The paper is organized as follows: In Section 2, we establish Theorem 1.1. Our
proof is based on the thorough comprehension of the corresponding linearized
operator of a model equation. It is also essential that we analyze the Fourier
series of some correction terms and prove its convergence. In Section 3, we
establish Theorem 1.2, and the key point of the proof is to use a radial
coefficient function and reduce all the iterations into radial cases. This
method made us avoid kernel functions in the linearized equation corresponding
to the quantized case.

##  2\. Non-quantized situation

In this section, we consider the non-quantized case. In other words, we set
$\alpha>-1$ and $\alpha\notin\mathbb{N}\cup\\{0\\}$. It is known that the
spherical Harnack holds around the origin when $\alpha$ is not an integer (See
[16]).

Denote $\lambda_{k}=u_{k}(0)$ and
$\epsilon_{k}=e^{-\frac{\lambda_{k}}{2(1+\alpha)}}$. Let $v_{k}$ be the
scaling of $u_{k}$:

$v_{k}(y)=u_{k}(\epsilon_{k}y)+2(1+\alpha)\log\epsilon_{k},\quad
y\in\Omega_{k}:=B(0,\epsilon_{k}^{-1}).$

Clearly, we need to construct $v_{k}$ to satisfy

(2.1) $\left\\{\begin{array}[]{lcl}\Delta
v_{k}(y)+|y|^{2\alpha}h_{k}(\epsilon_{k}y)e^{v_{k}(y)}=0,&&{\rm in}\
\,\Omega_{k},\\\ v_{k}(0)=0,\\\ |v_{k}(y_{1})-v_{k}(y_{2})|\leq C,&&{\rm for\
\ any}\ \,y_{1},y_{2}\in\partial\Omega_{k},\\\
v_{k}(y)\to-2\log(1+|y|^{2+2\alpha}),&&{\rm in}\
\,C_{loc}^{\beta}(\mathbb{R}^{2})\end{array}\right.$

where $\beta\in(0,1)$. It suffices to construct $\\{v_{k}\\}$ satisfying
(2.1). Since we can choose $h_{k}$ we require $h_{k}(0)=8(1+\alpha)^{2}$ for
convenience. Let

$U_{k}(y)=-2\log(1+|y|^{2+2\alpha})$

be a standard bubble that satisfies

(2.2) $\Delta U_{k}(y)+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}(y)}=0\quad{\rm
in}\ \,\mathbb{R}^{2}.$

Here we note that a uniform estimate of Bartolucci-Chen-Lin-Tarantello [1]
assures that any blow-up solution $v_{k}$ of (2.1) satisfies

$|v_{k}(y)-U_{k}(y)|\leq C,\quad y\in\Omega_{k}.$

We will construct our solutions based on the expansion of $v_{k}$ established
in [30]. Firstly, let us recall some notations and results in [30]. Denote

$\begin{split}g_{k}(r)&=-\frac{1}{4\alpha(1+\alpha)}\frac{r}{1+r^{2+2\alpha}},\quad
r=|y|,\\\
c_{1}^{k}(y)&=g_{k}(r)\epsilon_{k}\sum_{j=1}^{2}\partial_{j}h_{k}(0)\theta_{j},\quad\theta_{j}=\frac{y_{j}}{r}\
\ (j=1,2).\end{split}$

Then $c_{1}^{k}$ satisfies

(2.3) $\Delta
c_{1}^{k}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}(y)}c_{1}^{k}=-\sum_{j=1}^{2}\epsilon_{k}\partial_{j}h_{k}(0)y_{j}|y|^{2\alpha}e^{U_{k}(y)}\quad{\rm
in}\ \,\Omega_{k}.$

[30] tells us that $c_{1}^{k}$ is the second term in the expansion of $v_{k}$
if $\alpha>0$ is a non-integer. For the case $-1<\alpha<0$, Bartolucci-Yang-
Zhang [4] have established the same result. Here we point out that the radial
part of $c_{1}^{k}$ decays like $\epsilon_{k}r^{-1-2\alpha}$ at infinity. In
particular, for $r=\epsilon_{k}^{-1}$, the angular part of the function is
comparable to $\epsilon_{k}^{2+2\alpha}e^{i\theta}$, which means this term
contributes no oscillation on the boundary. So as long as $|\nabla\log
h_{k}(0)|\geq 2c>0$, we have $|\nabla\log h_{k}(0)+\nabla\psi_{k}(0)|\geq c$.

For the convenience of the readers, we comment that the construction of
$c_{1}^{k}$ is essentially solving

(2.4)
$\frac{d^{2}}{dr^{2}}g+\frac{1}{r}\frac{d}{dr}g+\Big{(}8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}-\frac{l^{2}}{r^{2}}\Big{)}g=-r^{1+2\alpha}e^{U_{k}},\quad
r>0,$

with $l=1$. From the proof of Lemma 2.1 in [30], we know two fundamental
solutions $F_{1}$ and $F_{2}$ of the homogeneous equation of (2.4) can be
written explicitly as follows:

(2.5)
$\begin{split}F_{1}(r)&=\frac{(\frac{l}{1+\alpha}+1)r^{l}+(\frac{l}{1+\alpha}-1)r^{l+2(1+\alpha)}}{1+r^{2(1+\alpha)}},\\\
F_{2}(r)&=\frac{(\frac{l}{1+\alpha}+1)r^{-l+2(1+\alpha)}+(\frac{l}{1+\alpha}-1)r^{-l}}{1+r^{2(1+\alpha)}}.\end{split}$

Therefore, we can verify that $g$ can be explicitly written with two
fundamental solutions above by the standard ODE methods.

For the motivation of adding more terms in the correction, we use the decay of
$c_{1}^{k}$ to obtain

(2.6)
$\displaystyle\Delta(U_{k}+c_{1}^{k})+8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}+c_{1}^{k}}$
$\displaystyle=$ $\displaystyle\Delta U_{k}+\Delta
c_{1}^{k}+8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}\big{(}1+c_{1}^{k}+\frac{(c_{1}^{k})^{2}}{2}\big{)}+O(\epsilon_{k}^{3})(1+r)^{-7-8\alpha}$
$\displaystyle=$ $\displaystyle
8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}\frac{(c_{1}^{k})^{2}}{2}-\epsilon_{k}\sum_{j}\partial_{j}h_{k}(0)\theta_{j}r^{1+2\alpha}e^{U_{k}}+O(\epsilon_{k}^{3})(1+r)^{-7-8\alpha}.$

At this moment we write the expansion of $h_{k}(\epsilon_{k}y)$:

(2.7) $\begin{split}h_{k}(\epsilon_{k}y)=&8(1+\alpha)^{2}+\epsilon_{k}\nabla
h_{k}(0)\cdot
y+\frac{\epsilon_{k}^{2}}{2}\partial_{11}h_{k}(0)(y_{1}^{2}-\frac{|y|^{2}}{2})+\frac{\epsilon_{k}^{2}}{2}\partial_{22}h_{k}(0)\cdot\\\
&(y_{2}^{2}-\frac{|y|^{2}}{2})+\epsilon_{k}^{2}\partial_{12}h_{k}(0)y_{1}y_{2}+\frac{\epsilon_{k}^{2}}{4}\Delta
h_{k}(0)|y|^{2}+O(\epsilon_{k}^{3})|y|^{3}\\\
=&8(1+\alpha)^{2}+\epsilon_{k}\nabla h_{k}(0)\cdot
y+\epsilon_{k}^{2}r^{2}\Theta_{2}+\frac{1}{4}\epsilon_{k}^{2}r^{2}\Delta
h_{k}(0)+O(\epsilon_{k}^{3})r^{3}.\end{split}$

where

$\begin{split}\Theta_{2}:=&\frac{1}{2}\partial_{11}h_{k}(0)(\theta_{1}^{2}-\frac{1}{2})+\partial_{12}h_{k}(0)\theta_{1}\theta_{2}+\frac{1}{2}\partial_{22}h_{k}(0)(\theta_{2}^{2}-\frac{1}{2})\\\
=&\frac{1}{4}(\partial_{11}h_{k}(0)-\partial_{22}h_{k}(0))\cos
2\theta+\frac{1}{2}\partial_{12}h_{k}(0)\sin 2\theta.\end{split}$

Based on (2.3), (2.6) and (2.7) we have

(2.8)
$\displaystyle\begin{split}&\Delta(U_{k}+c_{1}^{k})+h_{k}(\epsilon_{k}y)|y|^{2\alpha}e^{U_{k}+c_{1}^{k}}\\\
=&r^{2\alpha}e^{U_{k}}\Big{(}8(1+\alpha)^{2}\frac{(c_{1}^{k})^{2}}{2}+\Theta_{2}\epsilon_{k}^{2}r^{2}+\frac{1}{4}\epsilon_{k}^{2}r^{2}\Delta
h_{k}(0)+\epsilon_{k}\nabla h_{k}(0)\cdot yc_{1}^{k}\\\
&+O(\epsilon_{k}^{3}(1+r)^{3})\Big{)}.\end{split}$

Now we compute $(c_{1}^{k})^{2}$:

$\displaystyle(c_{1}^{k})^{2}=$
$\displaystyle\epsilon_{k}^{2}g_{k}^{2}(\partial_{1}h_{k}(0)\cos\theta+\partial_{2}h_{k}(0)\sin\theta)^{2}$
$\displaystyle=$ $\displaystyle\epsilon_{k}^{2}g_{k}^{2}\Big{(}\frac{|\nabla
h_{k}(0)|^{2}}{2}+\frac{1}{2}((\partial_{1}h_{k}(0))^{2}-(\partial_{2}h_{k}(0))^{2})\cos
2\theta$ $\displaystyle+\partial_{1}h_{k}(0)\partial_{2}h_{k}(0)\sin
2\theta\Big{)}$

Also the remaining term of the order $O(\epsilon_{k}^{2})$ is

$\displaystyle\epsilon_{k}\nabla h_{k}(0)\cdot yc_{1}^{k}$ $\displaystyle=$
$\displaystyle\epsilon_{k}^{2}g_{k}r(\partial_{1}h_{k}(0)\cos\theta+\partial_{2}h_{k}(0)\sin\theta)^{2}$
$\displaystyle=$ $\displaystyle\epsilon_{k}^{2}g_{k}r\Big{(}\frac{|\nabla
h_{k}(0)|^{2}}{2}+\frac{1}{2}(\partial_{1}h_{k}(0)^{2}-\partial_{2}h_{k}(0)^{2})\cos
2\theta+\partial_{1}h_{k}(0)\partial_{2}h_{k}(0)\sin 2\theta\Big{)}.$

To get rid of the terms with $e^{2i\theta}$ of the order $O(\epsilon_{k}^{2})$
in (2.8) we let $c_{2}^{k}$ be the solution of

$\displaystyle\Delta
c_{2}^{k}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}}c_{2}^{k}=-r^{2\alpha}e^{U_{k}}\Big{(}\epsilon_{k}^{2}r^{2}\Theta_{2}+\mathcal{A}(\frac{(c_{1}^{k})^{2}}{2}+\epsilon_{k}\nabla
h_{k}(0)\cdot yc_{1}^{k})\Big{)}$ $\displaystyle=$
$\displaystyle-\epsilon_{k}^{2}|y|^{2\alpha}e^{U_{k}}\Big{(}r^{2}\Theta_{2}+\big{(}4(1+\alpha)^{2}g_{k}^{2}+g_{k}r\big{)}\cdot((\partial_{1}h_{k}(0))^{2}-(\partial_{2}h_{k}(0))^{2})\frac{\cos
2\theta}{2}$ $\displaystyle+\partial_{1}h_{k}(0)\partial_{2}h_{k}(0)\sin
2\theta)\Big{)}.$

Note that $\mathcal{A}(\cdot)$ means the non-radial part of the term in the
parenthesis.

Since each term in $c_{2}^{k}$ is a product of a radial function and a
spherical harmonic function, we set $w_{1}$ to be a solution of

$\frac{d^{2}}{dr^{2}}w_{1}+\frac{1}{r}\frac{d}{dr}w_{1}+\Big{(}8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}-\frac{4}{r^{2}}\Big{)}w_{1}=r^{2+2\alpha}e^{U_{k}}$

with the control of $|w_{1}(r)|\leq C$ for all $r$. Similarly, we set $w_{2}$
to be a solution of

$\frac{d^{2}}{dr^{2}}w_{2}+\frac{1}{r}\frac{d}{dr}w_{2}+\Big{(}8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}-\frac{4}{r^{2}}\Big{)}w_{2}=r^{2\alpha}e^{U_{k}}\big{(}4(1+\alpha)^{2}g_{k}^{2}+g_{k}r\big{)}$

with $|w_{2}(r)|\leq C$ for all $r$. Two fundamental solutions of the
corresponding homogeneous equation can be seen in (2.5) with $l=2$.
Furthermore, we observe that the non-homogenous terms have good decay rates at
infinity. Therefore, the construction of $w_{1}$ and $w_{2}$ is standard. At
this point, it is easy to verify that $c_{2}^{k}$ can be constructed as

$\displaystyle c_{2}^{k}(y)=$
$\displaystyle\epsilon_{k}^{2}\Big{(}w_{1}(r)\Theta_{2}+w_{2}(r)\big{(}(\partial_{1}h_{k}(0))^{2}-(\partial_{2}h_{k}(0))^{2})\frac{\cos
2\theta}{2}$ $\displaystyle+\partial_{1}h_{k}(0)\partial_{2}h_{k}(0)\sin
2\theta\big{)}\Big{)}.$

Finally we use $c_{0}^{k}$ to handle the radial term of the order
$O(\epsilon_{k}^{2})$: We let $c_{0}^{k}$ solve

$\displaystyle\Delta c_{0}^{k}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}}c_{0}^{k}$
$\displaystyle=$
$\displaystyle-\epsilon_{k}^{2}|y|^{2\alpha}e^{U_{k}}\Big{(}r^{2}\frac{\Delta
h_{k}(0)}{4}+\frac{1}{2}|\nabla
h_{k}(0)|^{2}\big{(}4(1+\alpha)^{2}g_{k}(r)^{2}+g_{k}(r)r\big{)}\Big{)}.$

Since both $U_{k}$ and the right-hand side of the above are radial, we can
construct $c_{0}^{k}$ as a radial function $c_{0}^{k}(r)$ that satisfies

$\left\\{\begin{array}[]{ll}\frac{d^{2}}{dr^{2}}c_{0}^{k}(r)+\frac{d}{dr}c_{0}^{k}(r)+8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}c_{0}^{k}(r)\\\
\quad=-\epsilon_{k}^{2}r^{2\alpha}e^{U_{k}}\Big{(}\frac{r^{2}}{4}\Delta
h_{k}(0)+\frac{1}{2}|\nabla
h_{k}(0)|^{2}\big{(}4(1+\alpha)^{2}g_{k}(r)^{2}+g_{k}(r)r\big{)}\Big{)}.\\\
c_{0}^{k}(0)=\frac{d}{dr}c_{0}^{k}(0)=0.\end{array}\right.$

We only need to define $c_{0}^{k}$ for $0<r<\epsilon_{k}^{-1}$. It is easy to
use the standard ODE method to obtain

(2.9) $|c_{0}^{k}(r)|\leq C\epsilon_{k}^{2}(1+r)^{-2\alpha}\log(2+r),\quad
0<r<\epsilon_{k}^{-1}.$

Set $c_{k}=c_{0}^{k}+c_{1}^{k}+c_{2}^{k}$, we verify by direct computation
that

(2.10)
$\Delta(U_{k}+c_{k})+|y|^{2\alpha}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}}=E_{k}$
(2.11) $|E_{k}(y)|\leq c_{1}\epsilon_{k}^{3}(1+|y|)^{-1-2\alpha},\quad
y\in\Omega_{k}.$

So in order to find a solution with a non-vanishing coefficient, we need to
find $d_{k}$ to satisfy

(2.12)
$\Delta(U_{k}+c_{k}+d_{k})+|y|^{2\alpha}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}+d_{k}}=0,\quad\mbox{in}\quad\Omega_{k}.$

The difference between (2.10) and (2.12) gives

(2.13) $\Delta
d_{k}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}}d_{k}=-E_{k}-f(d_{k}).$

where

(2.14)
$f(d_{k})=-|y|^{2\alpha}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}}(e^{d_{k}}-1-d_{k})+|y|^{2\alpha}e^{U_{k}}(h_{k}(\epsilon_{k}y)e^{c_{k}}-h_{k}(0))d_{k}.$

is of higher order. Based on (2.13) we design an iteration scheme: Let
$d_{k}^{(0)}\equiv 0$ and $d_{k}^{(1)}$ satisfy

$\Delta
d_{k}^{(1)}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}}d_{k}^{(1)}=-E_{k}-f(d_{k}^{(0)}).$

In general we shall construct $d_{k}^{(m+1)}$ that satisfies

$\Delta
d_{k}^{(m+1)}+8(1+\alpha)^{2}|y|^{2\alpha}e^{U_{k}}d_{k}^{(m+1)}=-E_{k}-f(d_{k}^{(m)})$

and

(2.15) $d_{k}^{(m+1)}(0)=|\nabla d_{k}^{(m+1)}(0)|=0.$

Here we claim that there exists $c_{0}>0$ independent of $m$ and $k$ such that

(2.16) $|d_{k}^{(m)}(y)|\leq
c_{0}\epsilon_{k}^{3}(1+|y|)^{1-2\alpha}\log(2+|y|).$

The constant $c_{0}$ will be determined based on $c_{1}$ later. To prove this
uniform bound, we assume that (2.16) holds for $d_{k}^{(m)}$, and we shall
show that it also holds for $d_{k}^{(m+1)}$.

The projection to $1$ is the following equation: Let $f_{0}$ be the projection
of $d_{k}^{(m+1)}$ onto $1$, then $f_{0}$ solves

$\left\\{\begin{array}[]{ll}f_{0}^{\prime\prime}(r)+\frac{1}{r}f_{0}^{\prime}(r)+8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}f_{0}(r)+E_{0}^{k}=0,\quad
0<r\leq\epsilon_{k}^{-1}\\\
f_{0}(0)=\frac{\mathrm{d}}{\mathrm{d}r}f_{0}(0)=0,\end{array}\right.$

where $E_{0}^{k}$ is the corresponding projection of $E_{k}+f(d_{k}^{(m)})$
onto 1, and satisfies a similar bound of $E_{k}$:

(2.17) $|E_{0}^{k}(r)|\leq 2c_{1}\epsilon_{k}^{3}(1+|y|)^{-1-2\alpha}.$

The reason that $E_{0}^{k}$ has a worse coefficient $2c_{1}$ is that the
$d_{k}^{(m)}$ terms are absorbed.

We denote the two fundamental solutions of the homogeneous equation of $f_{0}$
as $u_{1}$ and $u_{2}$, where

$u_{1}(r)=\frac{1-r^{2+2\alpha}}{1+r^{2+2\alpha}},$

and $u_{2}(r)$ is comparable to $\log r$ near $0$ and infinity. Based on
standard ODE theory,

$f_{0}(r)=-u_{1}(r)\int_{0}^{r}tE_{1}^{k}(t)u_{2}(t)\mathrm{d}t-u_{2}(r)\int_{0}^{r}tE_{1}^{k}(t)u_{1}(t)\mathrm{d}t.$

Integrating the identity above, we know that

$f_{0}(r)=O(\epsilon_{k}^{3}(1+r)^{1-2\alpha}r|\log(r)|),\ \ \mathrm{at}\ \
0,\quad f_{0}(r)=O(\epsilon_{k}^{3}(1+r)^{1-2\alpha}\log(r)),\ \ \mathrm{at}\
\ \infty.$

In other words, we have the following estimate for $f_{0}$

$|f_{0}(r)|\leq c_{0}\epsilon_{k}^{3}(1+r)^{1-2\alpha}\log(2+r),$

where $c_{0}$ is a constant independent of $l$ and only depends on $c_{1}$.

Next, we consider the projections on high frequencies. For
$l\in\mathbb{N}^{+}$, let $f_{l}$ satisfies

$\left\\{\begin{array}[]{ll}f_{l}^{\prime\prime}(r)+\frac{1}{r}f_{l}^{\prime}(r)+\Big{(}8(1+\alpha)^{2}r^{2\alpha}e^{U_{k}}-\frac{l^{2}}{r^{2}}\Big{)}f_{l}(r)+E_{2,k}^{l}=0,\quad
0<r\leq\epsilon_{k}^{-1}\\\ f_{l}(0)=0.\end{array}\right.$

Here $E_{2,k}^{l}$ ($l\geq 1$) is the radial part of the projection of some
error term on $\cos({l\theta}$):

$E_{2,k}^{l}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}E_{k}\cos{(l\theta)}\mathrm{d}\theta$

The estimate of $E_{2,k}^{l}$ is

(2.18) $|E_{2,k}^{l}(r)|\leq 2c_{1}\epsilon_{k}^{3}(1+r)^{-1-2\alpha}$

In order to find $f_{l}$ we use two fundamental solutions $F_{1}$ and $F_{2}$
of the homogeneous equation, whose explicit expressions can be seen in (2.5).
As one can see that $F_{1}$ is comparable to $r^{l}$ at the origin and at
infinity, and $F_{2}$ is comparable to $r^{-l}$ at the origin and infinity. At
this point, we can construct $f_{l}$ as follows

$f_{l}(r)=-F_{1}(r)\int_{r}^{\infty}\frac{t}{2l}E_{2,k}^{l}(t)F_{2}(t)\mathrm{d}t-F_{2}(r)\int_{0}^{r}\frac{t}{2l}E_{2,k}^{l}(t)F_{1}(t)\mathrm{d}t.$

Integrate the identity above, we know that

$|f_{l}(r)|\leq\frac{c_{2}}{l^{2}}\epsilon_{k}^{3}(1+r)^{1-2\alpha},$

where $c_{2}$ is a constant independent of $l$. It is easy to see that
$f_{l}(0)=0$. Furthermore the summation of projections on all
$\cos{(l\theta})$ $(l\geq 1)$ is convergent. That is

$\big{|}\sum_{l\geq 1}f_{l}(r)\big{|}\leq
c_{2}\epsilon_{k}^{3}(1+r)^{1-2\alpha}\sum_{l\geq 1}\frac{1}{l^{2}}\leq
c_{2}\epsilon_{k}^{3}(1+r)^{1-2\alpha}$

In the same way we can construct the projection on $\sin{(l\theta)}$ for all
$l\geq 1$, called $\tilde{f}_{l}$, and the summation of $\tilde{f}_{l}$ is
convergent as well. $d_{k}^{(m)}$ is well-defined and satisfies the estimate
(2.16).

Thus by Brower fixed point theorem, we obtain the existence of $d_{k}$. The
construction is complete in this case.

The Laplacian term is also obviously true, which can be seen in the
construction. The construction of a non-quantized case is complete.

##  3\. Quantized situation

Let $N$ be a positive natural number, our goal is to construct a sequence of
blow-up solutions $u_{k}$ such that

$\Delta u_{k}+|x|^{2N}h_{k}(x)e^{u_{k}(x)}=0,\quad\mbox{in}\quad B_{1}$

such that the spherical Harnack holds around the origin, the only blow-up
point in $B_{1}$ and $\Delta\log h_{k}(0)$ do not tend to zero. Here
$\psi_{k}$ is the set of harmonic functions that eliminate the oscillation of
$u_{k}$ on $\partial B_{1}$.

The main result of this section is to prove the following theorem.

######  Theorem 3.1.

For any $N\in\mathbb{N}$, there exists $h_{k}(x)$ satisfying (1.3) and a
sequence of blow-up solutions $u_{k}$ of (1.2)(1.4)(1.5) such that $u_{k}$ is
simple and $|\Delta(\log h_{k})(0)|\geq c$ for some $c>0$ independent of $k$.

Proof of Theorem 3.1: We set

$h_{k}(x)=8(N+1)^{2}+|x|^{2},\quad x\in B_{1}.$

Obviously

$\nabla\log h_{k}(0)=0,\quad\Delta(\log h_{k})(0)=\frac{\Delta
h_{k}(0)}{h_{k}(0)}=\frac{1}{4(1+N)^{2}}.$

Let $v_{k}$ be the scaling of $u_{k}$ according to the maximum of $u_{k}$: Let

$\epsilon_{k}=e^{-\frac{u_{k}(0)}{2(1+N)}}$

and

$v_{k}(y)=u_{k}(\epsilon_{k}y)+2(1+N)\log\epsilon_{k}.$

The equation for $v_{k}$ is

$\Delta v_{k}+(1+\epsilon_{k}^{2}|y|^{2})|y|^{2N}e^{v_{k}}=0.$

Our goal is to construct $v_{k}$ satisfying the equation above based on the
global solution $U_{k}$. The classification theorem of Prajapat-Tarantello
[21] gives the standard bubble of $\Delta U+8(N+1)^{2}|y|^{2N}e^{U}=0$:

$U(y)=\log\frac{\lambda}{\big{(}1+\lambda|y^{N+1}-\xi|^{2}\big{)}^{2}}$

where parameters $\lambda>0$ and $\xi\in\mathbb{C}$.

Setting $\lambda=1$ and $\xi=0$ in $U$, we use the radial $U_{k}(y)$:

$U_{k}(y)=\log\frac{1}{\big{(}1+|y|^{2N+2}\big{)}^{2}}.$

Here we note that $\partial_{\lambda}|_{\lambda=0}U$,
$\partial_{\xi}|_{\xi=0}U$ and $\partial_{\bar{\xi}}|_{\xi=0}U$ for a basis
for the linearized space.

$\displaystyle\partial_{\lambda}\big{|}_{\lambda=1}U$
$\displaystyle=\frac{1-r^{2N+2}}{1+r^{2N+2}},$
$\displaystyle\partial_{\xi}\big{|}_{\xi=0}U$
$\displaystyle=\frac{2r^{N+1}}{1+r^{2N+2}}e^{-i(N+1)\theta},$
$\displaystyle\partial_{\bar{\xi}}\big{|}_{\xi=0}U$
$\displaystyle=\frac{2r^{N+1}}{1+r^{2N+2}}e^{i(N+1)\theta}.$

Because of this, we see that corresponding to $N$ we have

$\frac{2r^{N+1}}{1+r^{2N+2}}\sin((N+1)\theta),\quad\frac{2r^{N+1}}{1+r^{2N+2}}\cos((N+1)\theta)$

in the kernel, this is the reason we only obtain the non-vanishing estimate
for $\Delta(\log h_{k})(0)$. It would be interesting to construct a simple
blowup sequence with non-vanishing first-order coefficients.

Based on the fact $h_{k}(\epsilon_{k}y)=8(N+1)^{2}+\epsilon_{k}^{2}|y|^{2}$
and the equation of $U_{k}$, we have

$\Delta
U_{k}+h_{k}(\epsilon_{k}y)|y|^{2N}e^{U_{k}}=\epsilon_{k}^{2}|y|^{2N+2}e^{U_{k}}.$

In order to deal with the right-hand side of the equation above, we let
$c_{k}$ solve

$\Delta
c_{k}+8(N+1)^{2}|y|^{2N}e^{U_{k}}c_{k}=-\epsilon_{k}^{2}|y|^{2N+2}e^{U_{k}}.$

Similar with $c_{0}^{k}$ in the non-quantized case, we can construct $c_{k}$
as a radial function $c_{k}(r)$ satisfying

$\left\\{\begin{array}[]{ll}\frac{d^{2}}{dr^{2}}c_{k}(r)+\frac{d}{dr}c_{k}(r)+8(N+1)^{2}r^{2N}e^{U_{k}}c_{k}(r)=-\epsilon_{k}^{2}r^{2N+2}e^{U_{k}},\quad
0<r<\epsilon_{k}^{-1}\\\ c_{k}(0)=\frac{d}{dr}c_{k}(0)=0.\end{array}\right.$

After the standard ODE method, we obtain the estimate as in (2.9):

(3.1) $|c_{k}(r)|\leq C\epsilon_{k}^{2}(1+r)^{-2N}\log(2+r),\quad
0<r<\epsilon_{k}^{-1}.$

Note that $e^{U_{k}+c_{k}}=e^{U_{k}}(1+c_{k}+O(\epsilon_{k}^{4}))$. By direct
computation, we obtain

(3.2) $\Delta(U_{k}+c_{k})+|y|^{2N}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}}=E_{k}.$

Here $E_{k}$ is radial and satisfies

(3.3) $|E_{k}(y)|\leq c\epsilon_{k}^{3}(1+|y|)^{-1-2N},\quad y\in\Omega_{k},$

where $c$ is a positive constant independent of $k$.

Then we set $v_{k}=U_{k}+c_{k}+b_{k}$. Removing the equation for $U_{k}$ and
$c_{k}$ we write the equation of $b_{k}$ as

(3.4) $\Delta
b_{k}+|y|^{2N}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}+b_{k}}-|y|^{2N}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}}=-E_{k},\quad|y|\leq\tau\epsilon_{k}^{-1}.$

The equation can be further written as

(3.5) $\Delta
b_{k}+8(N+1)^{2}|y|^{2N}e^{U_{k}}b_{k}=-E_{k}-\hat{f}(b_{k}),\quad|y|\leq\tau\epsilon_{k}^{-1}$

where

$\hat{f}(b_{k})=-|y|^{2N}h_{k}(\epsilon_{k}y)e^{U_{k}+c_{k}}(e^{b_{k}}-1-b_{k})+|y|^{2N}e^{U_{k}}(h_{k}(\epsilon_{k}y)e^{c_{k}}-h_{k}(0))b_{k}.$

Similar to the non-quantized case, we construct $b_{k}$ by iteration. Let
$b_{k}^{(m)}\equiv 0$ and $b_{k}^{(1)}$ satisfy

$\Delta
b_{k}^{(1)}+8(N+1)^{2}r^{2N}e^{U_{k}}b_{k}^{(1)}=-E_{k}-\hat{f}(b_{k}^{(0)}).$

In general we construct $b_{k}^{(m+1)}$ satisfying

$\Delta
b_{k}^{(m+1)}+8(N+1)^{2}r^{2N}e^{U_{k}}b_{k}^{(m+1)}=-E_{k}-\hat{f}(b_{k}^{(m)})$

and $b_{k}^{(m+1)}(0)=0$. Denote
$F_{k}^{m}(r)=-E_{k}(r)-\hat{f}(b_{k}^{(m)}(r))$. Then by the iteration method
as before if we set

$\frac{d^{2}}{dr^{2}}b_{k}^{(m+1)}+\frac{1}{r}\frac{d}{dr}b_{k}^{(m+1)}+8(N+1)^{2}r^{2N}e^{U_{k}}b_{k}^{(m+1)}=F_{k}^{m},\quad
0<r<\tau\epsilon_{k}^{-1},$

and

$b_{k}^{(m+1)}(0)=0.$

The homogeneous equation has two fundamental solutions, one is

$u_{1}=\frac{1-r^{2N+2}}{1+r^{2N+2}}.$

The second fundamental solution $u_{2}$ satisfies $|u_{2}(r)|\leq
C\log\frac{1}{r}$ near $0$ and $\infty$. We can construct $b_{k}^{(m+1)}$ from
$b_{k}^{(m)}$ as

$b_{k}^{(m+1)}(r)=u_{1}(r)\int_{0}^{r}tF_{k}^{m}(t)u_{2}(t){\rm
d}t+u_{2}(r)\int_{0}^{r}tF_{k}^{m}(t)u_{1}(t){\rm d}t.$

If $b_{k}^{(m)}(t)$ satisfies

$|b_{k}^{(m)}(t)|\leq C\epsilon_{k}^{2}\log(2+t),$

one can verify by direct computation that $b_{k}^{(m+1)}$ satisfies the same
bound. Thus by standard Brower fixed point theorem, there is a $b_{k}$ such
that

$\Delta b_{k}+8(N+1)^{2}r^{2N}e^{U_{k}}b_{k}=-E_{k}-\hat{f}(b_{k}).$

Theorem 3.1 is established. $\Box$

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