#  Kurdyka–Łojasiewicz functions and Mapping Cylinder Neighborhoods

Daniel Cibotaru  Departamento de Matemática, Universidade Federal do Ceará,
Fortaleza, Brazil<EMAIL_ADDRESS>and  Fernando Galaz-García  Department
of Mathematical Sciences, Durham University, United Kingdom.  fernando.galaz-
<EMAIL_ADDRESS>
###### Abstract.

Kurdyka–Łojasiewicz (KŁ) functions are real-valued functions characterized by
a differential inequality involving the norm of their gradient. This class of
functions is quite rich, containing objects as diverse as subanalytic,
transnormal or Morse functions. We prove that the zero locus of a
Kurdyka–Łojasiewicz function admits a mapping cylinder neighborhood. This
implies, in particular, that wildly embedded topological $2$-manifolds in
$3$-dimensional Euclidean space, such as Alexander horned spheres, do not
arise as the zero loci of KŁ functions.

###### Key words and phrases:

mapping cylinder, relative manifold, Kurdyka–Łojasiewicz function, Alexander
horned sphere

###### 2010 Mathematics Subject Classification:

57R99

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Gradient flows of KŁ functions
  4. 4 Existence of mapping cylinder neighborhoods
  5. 5 Remarks on the KŁ nondegeneracy condition

##  1\. Introduction

A basic result of differential topology says that any closed subset of a
smooth manifold $M$, no matter how complicated, can arise as the zero locus of
a smooth, nonnegative real-valued function on $M$. On the other hand, if $M$
is analytic, the zero loci of real analytic functions on $M$ form, of course,
a more restricted class of closed sets whose properties have been intensively
studied for more than a century. Surprisingly, topological restrictions on
these sets come from a differential inequality: Łojasiewicz proved in [14,
Proposition 1, p. 92] that a real analytic function
$f\colon\mathbb{R}^{n}\to\mathbb{R}$ satisfies, in a neighborhood of a point
$p\in f^{-1}(0)$, the gradient inequality

$|\nabla_{x}f|\geq C|f(x)|^{\theta}$ (1.1)

for some $C>0$ and $0<\theta<1$.

The class of functions that satisfy gradient inequalities similar to (1.1) is
quite larger than that of real analytic functions on $\mathbb{R}^{n}$. Kurdyka
provided a pivotal extension in [12], where he showed that real-valued
functions whose graph belongs to an o-minimal structure $\mathcal{M}$ on
$(\mathbb{R},+,\cdot)$ satisfy, for some constants $C,\rho>0$, the inequality

$|\nabla_{x}(\psi\circ f)|\geq C\text{ for all }~{}x\in f^{-1}(0,\rho).$ (1.2)

In (1.2), the function $\psi\colon[0,\rho)\rightarrow[0,\infty)$ is strictly
increasing, continuous, and $C^{1}$ on $(0,\rho)$, with $\psi(0)=0$. In fact,
$\psi$ can be chosen to belong to $\mathcal{M}$. Examples of functions
satisfying a gradient inequality such as (1.2) can easily be found beyond the
world of o-minimal structures. It is enough to recall that the distance
function to a closed, smooth submanifold of a Riemannian ambient manifold
(see, for example, [19]) has a gradient of norm $1$ in a neighborhood of the
submanifold. It is therefore unsurprising that functions satisfying such
inequalities, henceforth called _KŁ functions_ , have become the focus of
stimulating research in the past two decades. This was motivated also by the
interest coming from convex optimization, complexity theory, and neural
networks. The article by Bolte, Daniilidis, Ley, and Mazet [2], a benchmark on
the topic of KŁ functions, and the references therein, will give the reader a
broader view of their alternative characterizations and applications. The
recently found counterexample to the Thom Gradient Conjecture [5] for the
class of KŁ functions only increases the intrigue surrounding them.

Our original interest in KŁ functions stemmed from the topological properties
of their zero locus. In [12], Kurdyka proved that the zero locus of a KŁ
function is a strong deformation retract of a neighborhood using the natural
deformation induced by the negative gradient flow. Having a neighborhood which
is a strong deformation retract is the first step in proving stronger
“embedding” properties of the zero locus, like, for example, that the pair
$(M,f^{-1}(0))$ is a cofibration (see [24]). However, the cofibration property
alone is still not enough to eliminate what one would perceive as pathological
or wild embeddings of topologically nice spaces such as spheres. We therefore
prove something stronger.

######  Theorem A.

The zero locus of a Kurdyka–Łojasiewicz function has a mapping cylinder
neighborhood. Moreover, this neighborhood is forward-time invariant with
respect to the negative gradient flow.

The question of which closed subsets $Z$ of a topological space $M$ have a
mapping cylinder neighborhood (see Definition 2.3) was intensively explored by
topologists in the 1970s. In a celebrated result, obtained as an application
of the theory of maps completion (see [20, 21]), Quinn showed that if $Z$ is
an absolute neighborhood retract (ANR) which is locally $1$-connected
($1$-LC), and $M$ is a topological manifold of dimension at least $5$, then
$Z$ has a mapping cylinder neighborhood.

The context of Quinn’s results and of our note as well is that of relative
manifolds $(M,Z)$. This means that $M\setminus Z$ is a manifold (topological
in [20], of class at least $C^{2}$ for us). In contrast to [20], we do not
assume that either $M$ or $Z$ are ANRs, nor that $Z$ is $1$-LC. Thus, Theorem
A does not seem to follow directly from Quinn’s general results in dimension
$\geq 5$.

In his Ph.D. thesis [17], Nicholson proved that, given a $3$-manifold $M$ and
a closed subset $C$ which is a topological complex (i.e., $C$ is homeomorphic
to some locally finite simplicial complex), then $C$ is _tamely embedded_ in
$M$ (i.e., $C$ is a subcomplex of a triangulation of $M$) if and only if it
has a mapping cylinder neighborhood. If $C$ is not tamely embedded, we say
that its embedding is _wild_ and $C$ is _wildly embedded_. Putting together
Nicholson’s results and Theorem A we immediately get the following.

######  Corollary B.

No wildly embedded topological $2$-manifold in a $3$-manifold can be the zero
locus of a Kurdyka–Łojasiewicz function.

One notorious example of a wild embedding is that of an _Alexander horned
sphere_ [1], an embedding of $\mathbb{S}^{2}$ into $\mathbb{S}^{3}$ such that
at least one of the two connected components of the complement is not simply
connected. The existence of such embeddings was a consequential result in
topology, first exhibited by Alexander [1] almost a hundred years ago. For
further examples of wild $2$-spheres, such as the _Antoine, Fox–Artin,_ and
_Bing spheres_ , we refer the reader to [4]. An uncountable family of
nonequivalent Alexander horned sphere embeddings in the Sobolev class
$W^{1,n}$ was produced more recently by Hajłasz and Zhou in [8]. It is clear
that any wild embedding of $\mathbb{S}^{2}$ is not flat, i.e., when seen with
codomain $\mathbb{S}^{3}$, it is not equivalent to the embedding of the
equatorial sphere. By a result of Moise [16], in dimension $3$ an embedding is
flat if, and only if, it is tame.

The second (and last) part of our note aims to complement the results of [2]
in terms of alternative characterizations of a KŁ function in a neighborhood
of a noninterior point of its zero locus $Z$. Given the general context of the
results of [2], some type of semiconvexity condition is necessary there for a
good development of the theory of subdifferential evolution equations. As
opposed to [2], we do not assume on any condition resembling semiconvexity at
the zero locus $Z$ on the functions we consider. In counterpart, they are more
regular on $M\setminus Z$ and our context is finite dimensional, with local
compactness playing an important role.

Given a relative manifold $(M,Z)$ and a nonnegative, continuous function
$f\colon M\rightarrow[0,\infty)$ with $Z=f^{-1}(0)$ and $f$ of class $C^{1}$
on $M\setminus Z$, we will say that a point $p\in\partial Z$ is _simple
nondegenerate_ if it has a neighborhood $U_{p}$ with $\nabla_{x}f\neq 0$ for
all $x\in U_{p}\setminus Z$. If the inequality (1.2) holds locally around $p$,
we say that $p$ is _KŁ nondegenerate_ and, if the desingularization function
$\psi$ in (1.2) can be chosen to be absoltely continuous, rather than $C^{1}$,
we will say that $p\in\partial Z$ is _weakly nondegenerate_ (see Definitions
2.1 and 5.3 for more details). We obtain the following characterization.

######  Theorem C.

Let $f$ be a KŁ function of class $C^{1}$ with zero locus $Z$. If
$p\in\partial Z$ is a simple nondegenerate point, then the following
assertions are equivalent:

  1. (1)

The point $p\in\partial Z$ is KŁ nondegenerate,

  2. (2)

The point $p\in\partial Z$ is weakly KŁ nondegenerate,

  3. (3)

There exists a compact neighborhood $K_{p}\ni p$ such that the upper
semicontinuous function

$\alpha^{K}(t)=\displaystyle{\frac{1}{\displaystyle\inf_{x\in f^{-1}(t)\cap
K_{p}}|\nabla_{x}f|}}$

is in $L^{1}(0,\rho)$ for some $\rho>0$.

  4. (4)

There exists an open neighborhood $U\ni p$, a positive number $\rho>0$, and a
continuous function $a\colon(0,\rho]\rightarrow(0,\infty)$ such that
$a^{-1}\in L^{1}(0,\rho)$, and

$\displaystyle|\nabla_{x}f|\geq a(f(x))\text{ for all }x\in U\setminus Z.$
(1.3)

The equivalence of KŁ nondegeneracy with the integrability condition (3) in
Theorem C above appears also in [2, Theorems 18 and 20]. In a sense, our proof
of Theorem C, which does not use the subgradient curves, is a shortcut adapted
to the context of relative manifolds. One important lemma from [2] about
integrable majorants of upper semicontinuous functions appears prominently in
our proof as well.

We conclude this note with a brief study of what one can describe as the
“opposite” gradient inequality to (1.3), namely

$\displaystyle|\nabla f|\leq b(f),$ (1.4)

where $b$ is a nonnegative continuous function with
$\int_{0}^{\rho}b^{-1}=\infty$ for some $\rho>0$. We note that such an
equation holds in a neighborhood of a simple nondegenerate point $p\in\partial
Z$ if and only if

$\int_{0}^{\rho}\frac{1}{\displaystyle\sup_{x\in f^{-1}(t)\cap
K_{p}}|\nabla_{x}f|}~{}dt=\infty$

for some $\rho>0$ and a compact neighborhood $K_{p}$ of $p$.

A quick look at the one-dimensional case shows that there do not exist nonzero
functions that vanish at a point on the real line and satisfy (1.4). In
general, we were only able to conclude that the zero locus $Z$ of $f$ should
be quite “pathological”. More precisely, no point in the boundary of $Z$ can
be reached from its complement by a Lipschitz curve. It is unclear whether
such situation can occur while the simple nondegeneracy condition is still
satisfied. An alternative characterization of a gradient inequality of type
(1.4) is also presented making use of a “dual” version of Lemma 45 from [2].

Some major advances in the topology of smooth manifolds have been pushed by
the idea that critical points of smooth functions say something about the
change in topology. The follwing question, whose answer we believe to be
negative, is a timid attempt at an “inverse problem”: _The topology of the
zero locus might say something about the immediate presence of critical
points_.

######  Question.

Can a wild 2-sphere be the zero locus of a nonnegative $C^{2}$ function
$f\colon S^{3}\rightarrow[0,\infty)$ with only simple nondegenerate points?

Our article is organized as follows. In Section 2 we collect basic definitions
and examples, and set up the framework for the proof of Theorem A. We prove
this result in Section 4. Section 5 contains the proof of Theorem C, along
with further considerations on nondegenerate points of KŁ functions.

######  Acknowledgements.

The first named author would like to thank Vincent Grandjean, Aris Daniilidis,
and Luciano Mari for some useful discussions.

##  2\. Preliminaries

Let $(M,d)$ be a connected, locally compact metric space and let $Z\subset M$
be a nonempty, closed, proper subspace such that $M\setminus Z$ is a $C^{k}$
($k\geq 1$) manifold of dimension $n\geq 1$. Following Spanier [23, Ch. 6,
Sec. 2], we call the pair $(M,Z)$ a _relative $C^{k}$ manifold_.

We emphasize that $(M,d)$ need not be complete. We will require, however, that
the restriction of the metric $d$ to $M\setminus Z$ comes from a Riemannian
metric on this manifold.

The set $Z$ might have nonempty interior. Since $M$ is connected and $Z\subset
M$ is proper, closed, and nonempty, $\partial Z\neq\emptyset$. This ensures
that Definition 2.1 is not vacuous.

We will call a function $\alpha\colon(A_{1},B_{1})\rightarrow(A_{2},B_{2})$
between pairs of sets, i.e., $B_{i}\subset A_{i}$, a _relative_ function if
both $\alpha(B_{1})\subset B_{2}$ and $\alpha(A_{1}\setminus B_{1})\subset
A_{2}\setminus B_{2}$ hold. This implies that $\alpha^{-1}(B_{2})=B_{1}$ and
$\alpha^{-1}(A_{2}\setminus B_{2})=A_{1}\setminus B_{1}$. The $C^{l}$
morphisms in the category of relative $C^{k}$-manifolds with $l\leq k$
considered in this article will be continuous relative functions
$f\colon(M_{1},Z_{1})\rightarrow(M_{2},Z_{2})$ such that
$f\bigr{|}_{M_{1}\setminus Z_{1}}$ is of class $C^{l}$ .

######  Definition 2.1.

Let $(M,Z)$ be a relative $C^{1}$ manifold and let
$f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a relative $C^{1}$ function.
A point $p\in\partial Z$ is _KŁ-nondegenerate_ (with respect to $f$) if there
exists a triple $(\rho,U,\Psi)$ with $\rho>0$ a positive constant, $U\subset
M$ an open neighborhood of $p$, and
$\Psi\colon([0,\rho),\\{0\\})\rightarrow([0,\infty),\\{0\\})$ is a $C^{1}$
relative function satisfying the following conditions:

  * (KŁ1)

$\Psi^{\prime}(t)>0$, for $t\in(0,\rho)$;

  * (KŁ2)

$|\nabla_{x}(\Psi\circ f)|\geq 1$ for all $x\in f^{-1}((0,\rho))\cap U$.

The relative function $f$ is a _Kurdyka–Łojasiewicz function_ (or, for short,
a _KŁ function_) if every point $p\in\partial Z$ is KŁ-nondegenerate.

The function $f$ is a _global_ (or _uniform_) KŁ function if the triple
$(\rho,U,\Psi)$ can be chosen so that $U$ is a neighborhood of $\partial Z$.

A relative $C^{1}$ function $f\colon(M,Z)\rightarrow(\mathbb{R},\\{0\\})$, not
necessarily nonnegative, is a KŁ function if its absolute value $|f|$ is KŁ.

######  Example 2.2.

  1. (1)

Let $M_{o}\subset\mathbb{R}^{n}$ be a connected, embedded $C^{k}$-submanifold
of dimension $m<n$ with nonempty boundary $\partial M_{o}$. Clearly, $M_{o}$
and $\partial M_{o}$ are disjoint. Let $Z\subset\partial M_{o}$ be an open
subset with respect to the subspace topology on $\partial M_{o}$. Let
$M=M_{o}\cup Z$ and observe that $Z$ is closed in $M$. Moreover,
$Z=U\cap\partial M_{o}$, where $U$ is open in $\mathbb{R}^{n}$. The set
$U^{\prime}=U\cap(M_{o}\cup\partial M_{o})$ is locally compact, as it is the
intersection of an open and a closed subset. Since $U^{\prime}=U\cap M$ is
open in $M$, and $Z\subset U^{\prime}$, it follows that every point $p\in Z$
has a neighborhood in $M$ which is locally compact. This property is obviously
shared also by the points $p\in M_{o}$. It follows that $M$ is locally compact
and $(M,Z)$ is a relative manifold.

Observe that, if we let $Z\subset\partial M_{o}$ be closed in $\partial M_{o}$
rather than open, then $M_{o}\cup Z$ might not be locally compact. For
example, when $M_{o}$ is the open upper half plane in $\mathbb{R}^{2}$ and $Z$
is a compact segment on the $x$-axis.

  2. (2)

Let $U$ be an open and bounded subset of $\mathbb{R}^{n}$. By the
Kurdyka–Łojasiewicz Theorem ([12, Theorem 1]), if $f\colon
U\rightarrow\mathbb{R}$ is a positive differentiable tame function (with
respect to some o-minimal structure on $(\mathbb{R},+,\cdot)$ containing $U$),
then there exists $\Psi$ as in Definition 2.1 (cf. [5, Introduction]). In
particular, real analytic and subanalytic functions are KŁ.

  3. (3)

If $\psi\colon([0,\infty),\\{0\\})\rightarrow([0,\infty),\\{0\\})$ is a
relative $C^{1}$ function with $\psi^{\prime}>0$, then $\psi\circ f$ is a KŁ
function whenever $f$ is a KŁ function.

  4. (4)

Let $M$ be a Riemannian manifold (possibly with boundary) and let $Z\subset M$
be a smooth, properly embedded submanifold of $M$. Fix $p>0$ and, given $x\in
M$, let $f(x)=\tilde{d}(x,Z)^{p}$, where $d(\cdot,Z)$ is the distance function
to $Z$ and $\tilde{d}(\cdot,Z)$ is a function that coincides with $d$ on an
open neighborhood $U$ of $Z$ and is smooth and nonvanishing away from $Z$. One
takes $\Psi(t)=t^{1/p}$. It is well-known that $|\nabla d(\cdot,Z)|\equiv 1$
on some $U\setminus Z$. Clearly, if $p=1$, $f$ is just continuous at $Z$.

  5. (5)

nonnegative transnormal functions (see Wang [25]) are KŁ. These are functions
$f\colon M\rightarrow\mathbb{R}$ on a smooth manifold that satisfy
$|df|^{2}=b(f)$ for some smooth function $b$ (see Remark 5.7).

  6. (6)

Let $f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a relative $C^{1}$
function. Observe that a necessary condition for $f$ to be KŁ is that there
should exist a neighborhood $U$ of $Z$ such that $\nabla f$ does not vanish on
$U\setminus Z$. An example of a function $f$ on $\mathbb{R}^{2}$ with a
gradient orbit spiraling around the unit circle $S^{1}\subset\mathbb{R}^{2}$
which is a connected component of the zero locus $Z$ of $f$ may be found in
[18, p. 14]. The gradient of $f$ does not vanish on some $U\setminus S^{1}$
and $f$ cannot be KŁ, as the gradient orbit will have infinite length. The
zero locus $Z$ is disconnected for this $f$, with the two other connected
components of $Z$ spiraling around $S^{1}$. In [2], the authors gave an
example of a nonnegative $C^{2}$ convex function on $\mathbb{R}^{2}$ whose
minimum locus is $S^{1}$ and which is not KŁ. More recently, Bolte and Pauwels
[3] constructed convex non-KŁ functions with regularity $C^{k}$ $(k\geq 1)$.

  7. (7)

Let $f\colon M\rightarrow\mathbb{R}$ be a Morse function on a smooth
$n$-dimensional Riemannian manifold $M$ and suppose that $0$ is a critical
value. Then $f^{+}=\max\\{f,0\\}$ is a KŁ function with zero locus
$Z=f^{-1}((-\infty,0])$. Note that $f^{+}$ is obviously KŁ at a point
$p\in\partial Z=f^{-1}(0)$ which is noncritical for $f$. If $p\in\partial Z$
is a critical point of $f$, one may use the Morse lemma [15] to produce a
coordinate chart $\varphi\colon V\subset M\to U\subset\mathbb{R}^{n}$ with
$p\in V$ and such that $\varphi(p)=0\in U$. Letting
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$, the local expression for $f$ on $U$
is given by

$f(x)=\frac{1}{2}(x_{1}^{2}+\ldots+x_{k}^{2}-x_{k+1}^{2}-\ldots x_{n}^{2}).$

In these coordinates, the metric on $M$ is given by a smooth function $g\colon
U\rightarrow\operatorname{Sym}^{+}(\mathbb{R}^{n})$ into the positive
symmetric matrices $\operatorname{Sym}^{+}(\mathbb{R}^{n})$ with $g(0)=I_{n}$,
the $n\times n$ identity matrix. The gradient is then

$\displaystyle\nabla_{x}f$
$\displaystyle=g^{-1}(x_{1},\ldots,x_{k},-x_{k+1},\ldots,-x_{n})$
$\displaystyle=g^{-1}(Jx),$

where $J=\left(\begin{array}[]{cc}I_{k}&0\\\ 0&-I_{n-k}\end{array}\right)$.
Hence, if we let $G\colon U\rightarrow\operatorname{Sym}^{+}(\mathbb{R}^{n})$
be given by

$G(x)=(g^{-1}J)^{T}(g^{-1}J),$

then

$|\nabla_{x}f|^{2}=\|x\|_{G(x)}^{2},$ (2.1)

where $\|\cdot\|_{G(x)}$ is the norm determined by $G(x)$. Since $G$ is
continuous and $G(0)=I_{n}$, there exists $W\subset U$ and $0<C<1$ such that
$\lambda_{G(x)}\geq C$ for all $\lambda\in\sigma(G(x))$, for all $x\in W$. It
follows then that

$\|x\|_{G(x)}^{2}\geq C\|x\|_{\operatorname{eucl}}^{2}\text{ for all }~{}x\in
W.$ (2.2)

We obviously have $\|x\|_{\operatorname{eucl}}^{2}\geq 2|f(x)|$, which
together with (2.1), (2.2) implies that

$|\nabla_{x}f^{+}|^{2}\geq 2Cf^{+}(x)$

for all $x\in W\setminus(f^{+})^{-1}(0)$. So $\Psi(t)=C^{\prime}t^{1/2}$ for
some $C^{\prime}>0$. A similar reasoning works if $f$ is a Morse–Bott
function.

### Mapping cylinder neighborhoods.

Let $f\colon X\to Y$ be a continuous function between topological spaces.
Recall that the _mapping cylinder_ $M_{f}$ of $f$ is the quotient space of the
disjoint union $(X\times[0,1])\sqcup Y$ obtained after identifying each
$(x,0)\in X\times[0,1]$ with $f(x)\in Y$. By identifying each $x\in X$ with
$(x,1)\in M_{f}$ we may consider $X$ and $Y$ as closed subsets of $M_{f}$. We
will denote the interior of an arbitrary set $A$ in a given topological space
by $\operatorname{int}(A)$. Let us now recall the following definition (cf.
[13, 17, 21]).

######  Definition 2.3.

Let $X$ be a topological space and let $C$ be a closed subset of $X$. A closed
neighborhood $N\supset C$ is a _mapping cylinder neighborhood_ of $C$ if there
exists a surjective function $f\colon\partial N\rightarrow\partial C$ and a
homeomorphism $\varphi\colon N\setminus{\kern 0.0ptC}^{\mathrm{o}}\rightarrow
M_{f}$ such that

$\varphi(p)=\left\\{\begin{array}[]{cc}[(1,p)],&\mbox{if}\;p\in\partial
N;\\\\[7.0pt] \;\quad\ \,[p],&\mbox{if}\;p\in\partial C.\end{array}\right.$

Let $(M,Z)$ be a relative $C^{k}$ manifold ($k\geq 2$). A mapping cylinder
neighborhood $N\supset Z$ is _$C^{l}$ regular_, with $l\leq k$, if $\partial
N$ is a $C^{l}$ hypersurface in $M\setminus Z$.

######  Remark 2.4.

The homeomorphism $\varphi\colon N\setminus\operatorname{int}(C)\to M_{f}$ in
Definition 2.3 extends to a homeomorphism $\widetilde{\varphi}\colon
N\rightarrow M_{\tilde{f}}$, where $\tilde{f}$ is just $f$ with codomain $C$,
by letting $\widetilde{\varphi}$ be the identity on $\operatorname{int}(C)$.
We will therefore not distinguish between a mapping cylinder neighborhood of
$C$ and a mapping cylinder neighborhood of $\partial C$.

##  3\. Gradient flows of KŁ functions

We start this section by providing an alternative approach (in Proposition
3.4) to proving the uniform boundedness of the lengths of gradient
trajectories of a KŁ function (see [12, Theorem 2] or [2, Theorem 18]). Recall
first the standard theorem on dependence on initial conditions for autonomous
(i.e., time independent) ordinary differential equations (cf. [9, Ch. 7.3]).

######  Theorem 3.1.

Let $X\colon\Omega\rightarrow\mathbb{R}^{n}$ be a $C^{1}$ vector field on an
open set $\Omega\subset\mathbb{R}^{n}$ and let $x_{0}\in\Omega$. Let
$\varphi_{x_{0}}$ be the unique solution of the initial value problem

$\varphi_{x_{0}}^{\prime}=X(\varphi_{x_{0}}),\qquad\varphi_{x_{0}}(0)=x_{0},$
(3.1)

defined on its maximal interval $(\omega^{-}_{x_{0}},\omega^{+}_{x_{0}})$.
Then the set

$D=\\{(t,x_{0})\in\mathbb{R}\times\Omega\mid
x_{0}\in\Omega,\;t\in(\omega^{-}_{x_{0}},\omega^{+}_{x_{0}})\\}$

is open and $\varphi\colon D\rightarrow\Omega$, defined by
$\varphi(t,x_{0})=\varphi_{x_{0}}(t)$, is a $C^{1}$ function.

It is natural to ask if anything can be inferred about the continuity of the
functions

$\displaystyle\omega^{-}\colon\Omega\rightarrow[-\infty,0),$
$\displaystyle\quad x_{0}\mapsto\omega^{-}_{x_{0}};$
$\displaystyle\omega^{+}\colon\Omega\rightarrow(0,\infty],$
$\displaystyle\quad x_{0}\mapsto\omega^{+}_{x_{0}}.$

Owing to the classical alternative characterization of upper and lower
semicontinuity involving the epigraph of a function, the following result
follows immediately from Theorem 3.1.

######  Corollary 3.2.

The function $\omega^{-}$ is upper semicontinuous and the function
$\omega^{+}$ is lower semicontinuous.

One cannot expect a better regularity than Corollary 3.2, as the example of an
open set in $\mathbb{R}^{2}$ contained between the graphs of a negative upper
semicontinuous function and a positive lower semicontinuous function with the
flow induced by $\partial_{y}$.

However, the following result provides a link between the left and right end
maps $\omega^{\mp}$ for one vector field and its product with a positive
function.

######  Lemma 3.3.

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, let
$X\colon\Omega\rightarrow\mathbb{R}^{n}$ be a $C^{1}$ vector field, and fix
$x_{0}\in\Omega$. Let $h\colon\Omega\rightarrow(0,\infty)$ be a positive,
$C^{1}$ function, set $Y=hX$, and consider the initial value problems

$\varphi_{x_{0}}^{\prime}=X(\varphi_{x_{0}}),\qquad\varphi_{x_{0}}(0)=x_{0}$
(3.2)

and

$\psi_{x_{0}}^{\prime}=Y(\psi_{x_{0}}),\qquad\psi_{x_{0}}(0)=x_{0}.$ (3.3)

Let $D_{1}\subset\mathbb{R}\times\Omega$ be the domain of definition of the
flow $\varphi$ of $X$ and let $D_{2}\subset\mathbb{R}\times\Omega$ be the
domain of definition of the flow $\psi$ of $Y$. Then there exists a $C^{1}$
diffeomorphism $\theta\colon D_{1}\rightarrow D_{2}$ making the following
diagrams commutative:

$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\pi_{2}}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\pi_{2}}$$\textstyle{\Omega}$
$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\varphi}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\psi}$$\textstyle{\Omega}$

where $\pi_{2}\colon\mathbb{R}\times\Omega\rightarrow\Omega$ is projection
onto the second factor. More precisely, for each $(t,x_{0})\in D_{1}$,

$\theta(t,x_{0})=(\theta_{x_{0}}(t),x_{0})$

with

$\displaystyle\theta_{x_{0}}(t)=\int_{0}^{t}\frac{1}{h(\varphi_{x_{0}}(r))}~{}dr.$
(3.4)

###### Proof.

Let
$\varphi_{x_{0}}\colon(\omega_{x_{0}}^{-},\omega_{x_{0}}^{+})\subset\mathbb{R}\to\Omega$
be the unique solution to the initial value problem (3.2). Then the unique
solution
$\psi_{x_{0}}\colon(\nu^{-}_{x_{0}},\nu_{x_{0}}^{+})\subset\mathbb{R}\to\Omega$
to the initial value problem (3.3) is given by
$\psi_{x_{0}}=\varphi_{x_{0}}\circ\gamma_{x_{0}}$, where
$\gamma_{x_{0}}\colon(\nu^{-}_{x_{0}},\nu_{x_{0}}^{+})\to(\omega_{x_{0}}^{-},\omega_{x_{0}}^{+})$
is the unique solution to the initial value problem on the real line given by

$\gamma_{x_{0}}^{\prime}(t)=(h\circ\varphi_{x_{0}})(\gamma_{x_{0}}(s)),\qquad\gamma_{x_{0}}(0)=0.$

Hence $\gamma_{x_{0}}=\theta^{-1}_{x_{0}}$, where

$\theta_{x_{0}}(t)=\int_{0}^{t}\frac{1}{h(\varphi_{x_{0}}(r))}~{}dr.$

Note that
$\theta_{x_{0}}\colon(\omega_{x_{0}}^{-},\omega_{x_{0}}^{+})\to\mathbb{R}$ is
a $C^{1}$ diffeomorphism onto its image due to the positivity of $h$. Thus, we
get thus a bijective correspondence between the two initial value problems
(3.2) and (3.3). Since the intervals $(\omega_{x_{0}}^{-},\omega_{x_{0}}^{+})$
and $(\nu_{x_{0}}^{-},\nu_{x_{0}}^{+})$ on which the solutions
$\varphi_{x_{0}}$ and $\psi_{x_{0}}$ are defined are maximal, it follows that
$\theta_{x_{0}}$ is in fact a $C^{1}$ diffeomorphism between
$(\omega_{x_{0}}^{-},\omega_{x_{0}}^{+})=D_{1}\cap(\mathbb{R}\times\\{x_{0}\\})$
and
$(\nu_{x_{0}}^{-},\nu_{x_{0}}^{+})=D_{2}\cap(\mathbb{R}\times\\{x_{0}\\}$).
The explicit expression for $\theta$ makes it clear that it is $C^{1}$ also in
the $x$ directions. It follows then that $\theta\colon D_{1}\to D_{2}$ is a
diffeomorphism with the desired properties. ∎

Let us illustrate how Lemma 3.3 can be of good use.

######  Proposition 3.4.

Let $(M,Z)$ be a relative $C^{k}$ manifold ($k\geq 2$) and let
$f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a relative $C^{2}$ function
such that $\nabla f\neq 0$ on $M\setminus Z$. Assume that

$\displaystyle|\nabla_{x}(\Psi\circ f)|\geq 1,\qquad x\in f^{-1}(D_{\Psi})\cap
M\setminus Z$ (3.5)

for some nonnegative, nondecreasing, absolutely continuous function $\Psi$,
where $D_{\Psi}$ is the set of differentiability points for $\Psi$. Then the
lengths of the integral curves $\alpha_{x}$ of $-\nabla f$ with initial
condition $x\in M\setminus Z$ satisfy

$\displaystyle\operatorname{Length}(\alpha_{x})\leq\Psi(f(x))\text{ for all
}x\in M\setminus Z.$ (3.6)

###### Proof.

Let $\varphi_{x_{0}}$ be an integral curve for $X:=-\frac{\nabla f}{|\nabla
f|^{2}}$ starting at $x_{0}$. Clearly, $\varphi_{x_{0}}$ is a
reparametrization of $\alpha_{x_{0}}$. It is well-known (see [15]) that the
trajectories for $X$ travel in $t$ units of time from $f$-level $a$ to
$f$-level $a-t$. Hence $f(\varphi_{x_{0}}(r))=f(x_{0})-r$. Setting $h=|\nabla
f|$, we get from (3.5) that

$\displaystyle|h(\varphi_{x_{0}}(r))|^{-1}$
$\displaystyle\leq|\Psi^{\prime}(f(\varphi_{x_{0}}(r)))|$ (3.7)
$\displaystyle=\Psi^{\prime}(f(x_{0})-r)$

for every $r\in\mathbb{R}$ such that $f(x_{0})-r\in D_{\Psi}$. Then, equation
(3.4) in Lemma 3.3 and the Fundamental Theorem of Calculus yield that

$\displaystyle\theta_{x_{0}}(t)$
$\displaystyle\leq\int_{0}^{t}\Psi^{\prime}(f(x_{0})-r)dr=\int_{f(x_{0})-t}^{f(x_{0})}\Psi^{\prime}(u)du$
$\displaystyle=\Psi(f(x_{0}))-\Psi(f(x_{0})-t)$ (3.8)

for all $t\leq\omega_{x_{0}}^{+}$. Clearly, $\omega_{x_{0}}^{+}\leq f(x_{0})$
since $f(\varphi_{x_{0}}(r))\geq 0$ for all $r\leq\omega_{x_{0}}^{+}$. Since
$\theta_{x_{0}}$ is an increasing function (3) implies that

$\displaystyle\nu_{x_{0}}^{+}$
$\displaystyle=\theta_{x_{0}}(\omega_{x_{0}}^{+})\leq\theta_{x_{0}}(f(x_{0}))\leq\Psi(f(x_{0})).$
(3.9)

Note that $|Y|=1$ for $Y:=hX$, hence $\nu^{+}_{x_{0}}$ is the length of the
curve $\alpha_{x_{0}}$, between the initial position until it leaves
$M\setminus Z$.

∎

######  Remark 3.5.

Kurdyka’s proof in [12] of the uniform boundedness of the lengths of the
trajectories is based on the Mean Value Theorem, which, in turn, requires
$\Psi$ to be (at least) $C^{1}$, whereas the proof of Proposition 3.4 requires
only absolute continuity from $\Psi$. However, as we will see in Section 5,
there is no real gain in weakening the regularity assumptions on $\Psi$ in
Definition 2.1.

The next result, inspired by Kurdyka’s Proposition 3 in [12], is the first
important ingredient for the proof of Theorem A.

######  Proposition 3.6.

Let $(M,Z)$ be a relative $C^{k}$ manifold $(k\geq 2)$, let
$f\colon(M,Z)\to([0,\infty),\\{0\\})$ be a $C^{2}$, global KŁ function with
associated triple $(\rho,U,\Psi)$. For every $x_{0}\in M\setminus Z$ let
$\alpha_{x_{0}}\colon(\mu_{x_{0}}^{-},\mu_{x_{0}}^{+})\rightarrow M\setminus
Z$ be the maximal solution of the Cauchy problem

$\displaystyle x^{\prime}$ $\displaystyle=-\nabla_{x}f,$ $\displaystyle x(0)$
$\displaystyle=x_{0}.$

Then the following assertions hold:

  1. (1)

There exists an open neighborhood $V\subset f^{-1}([0,\rho))$ of $Z$ such that
the integral curves of $-\nabla f$ originating at points in $V\setminus Z$
have forward-limit points on $\partial Z$.

  2. (2)

The forward-limit mapping

$\displaystyle\Lambda\colon V\setminus Z$ $\displaystyle\rightarrow\partial Z$
$\displaystyle x$
$\displaystyle\mapsto\displaystyle\lim_{t\rightarrow\mu^{+}_{x}}\alpha_{x}(t)$

induces a retraction $R\colon V\rightarrow Z$.

  3. (3)

The length function

$\displaystyle L\colon V\setminus Z$ $\displaystyle\rightarrow(0,\infty)$
$\displaystyle x$
$\displaystyle\mapsto\operatorname{Length}(\alpha_{x}\bigr{|}_{[0,\mu_{x}^{+})})$

is continuous.

###### Proof.

For (1), consider the triple $(\rho,U,\Psi)$ associated to the global KŁ
function $f\colon M\to[0,\infty)$ and set $g=\Psi\circ f$. Fix $x_{0}\in
U\setminus Z$ and let $L_{x_{0}}$ be the length of the curve
$\alpha_{x_{0}}\bigr{|}_{[0,\mu_{x_{0}}^{+})}$. We know from (3.6) that
$L_{x_{0}}\leq g(x_{0})$.

Let $\psi_{x_{0}}\colon[0,L_{x_{0}})\rightarrow M$ be the arc length
reparametrization of $\alpha_{x_{0}}\bigr{|}_{[0,\mu_{x_{0}}^{+})}$.

Let $\widehat{M}$ be the metric completion of $M$. Note that
$\psi_{x_{0}}\colon[0,L_{x_{0}})\rightarrow M\subset\widehat{M}$ is Lipschitz,
hence uniformly continuous. Since $L_{x_{0}}<\infty$, by a standard result,
the curve $\psi_{x_{0}}$ has a unique continuous extension

$\widehat{\psi}_{x_{0}}\colon[0,L_{x_{0}}]\rightarrow\widehat{M}.$

Since $M$ is locally compact, $M$ is open in $\widehat{M}$, and hence
$\partial M=\widehat{M}\setminus M$. Let $d(\cdot,\partial M)$ be the distance
to $\partial M$ and set $d(x,\partial M)=\infty$ if $\partial M=\varnothing$.
Consider now the set

$\displaystyle V=\left\\{x\in f^{-1}([0,\rho))\mid d(x,\partial
M)>{g(x)}\right\\}.$ (3.10)

Notice now that $V\subset M$ is open and contains $f^{-1}(0)=g^{-1}(0)$.
Indeed, since $M$ is locally compact, $\partial M$ is closed in $\widehat{M}$.
Therefore, if $x\in\widehat{M}$ then $d(x,\partial M)=0$ if and only if
$x\in\partial M$.

Let $x_{0}\in V\setminus f^{-1}(0)$. We show that
$\widehat{\psi}(L_{x_{0}})\in M$. Suppose that this is not the case, i.e.,
$\widehat{\psi}_{x_{0}}(L_{x_{0}})\in\partial M$. One gets a contradiction
from the following string of inequalities

$\displaystyle\operatorname{Length}(\psi_{x_{0}})$
$\displaystyle=\operatorname{Length}(\widehat{\psi}_{x_{0}})$
$\displaystyle\geq d(x_{0},\widehat{\psi}_{x_{0}}(L_{x_{0}}))$
$\displaystyle\geq d(x_{0},\partial M)$ $\displaystyle>g(x_{0})$
$\displaystyle\geq\operatorname{Length}(\psi_{x_{0}}).$

Hence $\widehat{\psi}_{x_{0}}(L_{x_{0}})\in M$ for every $x_{0}\in V\setminus
Z$. We deduce from the maximality of $\alpha_{x_{0}}$ that
$\widehat{\psi}_{x_{0}}(L_{x_{0}})\in Z=f^{-1}(0)=g^{-1}(0)$. This concludes
the proof of assertion (1).

With $V\subset M$ as in (3.10) above, consider the mapping $R\colon V\to Z$
given by

$R(x):=\begin{cases}\displaystyle\lim_{t\rightarrow\mu_{x}^{+}}\alpha_{x}(t),&x\in
V\setminus Z;\\\ x,&x\in Z.\end{cases}$

By the proof of item (1), $R$ is well-defined. To prove the continuity of $R$
we proceed as follows. Fix $x_{0}\in V$, let $\varepsilon>0$, and consider the
open ball in $Z$ centered at $R(x_{0})$ given by

$\displaystyle\Omega_{\varepsilon}=\\{x\in Z\mid
d(x,R(x_{0}))<\varepsilon\\}.$ (3.11)

If $x_{0}\in\operatorname{Int}(Z)$, there is nothing to prove. If
$x_{0}\in\partial Z$, let

$U=\\{x\in V\mid{g(x)}+d(x,x_{0})<\varepsilon\\}.$

Since $g$ is continuous, $U$ is an open subset of $M$ and contains $x_{0}$.
Then, for all $x\in U$,

$\displaystyle d(R(x_{0}),R(x))$ $\displaystyle=d(x_{0},R(x))$
$\displaystyle\leq d(x_{0},x)+d(x,R(x))$ $\displaystyle\leq
d(x_{0},x)+\operatorname{Length}(\alpha_{x})$ $\displaystyle\leq
d(x_{0},x)+g(x)$ $\displaystyle<\varepsilon.$

Hence, $R(x)\in\Omega_{\epsilon}$, proving that $R$ is continuous at
$x_{0}\in\partial Z$.

Finally, if $x_{0}\in V\setminus Z$ let
$\Omega_{\varepsilon}^{c}:=Z\setminus\Omega_{\varepsilon}$. Notice that the
function $h\colon V\to\mathbb{R}$ given by

$h(y)=d(y,\Omega_{\varepsilon}^{c})-{g(y)}$

is continuous and $h(R(x_{0}))>0$. Hence, one can find $x_{1}\in V\setminus Z$
close to $R(x_{0})$ and on the integral curve determined by $x_{0}$, along
with a sufficiently small ball $B_{1}\subset V\setminus Z$ around $x_{1}$ such
that

$\displaystyle B_{1}\subset\\{y\in V\mid
d(y,\Omega_{\varepsilon}^{c})>{g(y)}\\}.$ (3.12)

On the other hand, we have, for every $y\in B_{1}$,

$\displaystyle d(y,R(y))$ $\displaystyle\leq\operatorname{Length}(\alpha_{y})$
$\displaystyle\leq{g(y)}$ $\displaystyle<d(y,\Omega^{c}_{\varepsilon}).$

This can only mean that $R(y)\in\Omega_{\varepsilon}$ for all $y\in B_{1}$.
Now, given $B_{1}$, there exists an open ball $B_{0}$ around $x_{0}$ such that
all trajectories that start in $B_{0}$ will cross $B_{1}$. Hence the function
$R$ will take $B_{0}$ to $\Omega_{\epsilon}$. Thus, $R$ is continuous also on
$V\setminus Z$. Clearly, $R$ is a retraction. This concludes the proof of item
(2).

For item (3), we use the notation of Proposition 3.4. Observe that
$\omega_{x}^{+}=f(x)$ for every $x\in V\setminus Z$. Note that the length
function coincides on $V\setminus Z$ with the function

$\displaystyle\nu^{+}(x)$
$\displaystyle=\theta_{x}(\omega_{x}^{+})=\int_{0}^{f(x)}\frac{1}{h(\varphi_{x}(r))}~{}dr,$

where $h=|\nabla f|$. By inequality (3.7), we can use the Lebesgue Dominated
Convergence Theorem for the continuous family of functions
$H_{x}(r):=h(\varphi_{x}(r))^{-1}$ to conclude that $\nu^{+}$ is indeed
continuous. ∎

######  Remark 3.7.

If $M$ is a $C^{2}$ manifold to begin with and
$f\colon(M,Z)\to(\mathbb{R},\\{0\\})$ is a nonnegative $C^{1,1}$ function on
$M$, then $\nabla f$ exists, is continuous on $M$, and equals $0$ on the
minimal locus $f^{-1}(0)$. Therefore, if $f$ is a KŁ function, the negative
gradient flow is forward-complete in $V$. This is because a trajectory of
finite length will stay within a compact subset of the domain of definition of
the vector field $-\nabla f$, which is defined everywhere on $V$ and
continuous. Hence, the trajectory will exist for all $t>0$.

Examining the proof of Proposition 3.6, one notices that one can easily remove
the global KŁ condition from its hypotheses by taking $V=\bigcup_{p\in\partial
Z}V_{p}$, where $V_{p}$ is the neighborhood on which the retraction $R$ is
defined under a local KŁ nondegeneracy condition.

######  Corollary 3.8.

The statements of Proposition 3.6 hold for general KŁ functions.

##  4\. Existence of mapping cylinder neighborhoods

Proposition 3.6 is the main step in the natural generalization of Kurdyka’s
strong deformation retract result [12, Proposition 3] to the class of KŁ
functions. In fact, one could have easily gone an extra step already in
Proposition 3.6 in order to recover the full statement from [12]. However,
since this is a direct corollary to Theorem A, we chose to skip it. One
comment is in order. One consequence of the fact $Z$ has a mapping cylinder
neighborhood is that the pair $(M,Z)$ is an NDR (neighborhood deformation
retract) pair (see [6, Definition 6.2]), or equivalently a closed cofibration
(provided the space under consideration is compactly generated, for example,
Hausdorff and locally compact). One then could ask the question of whether
wildly embedded submanifolds, such as the Alexander horned sphere, could be
excluded as the zero locus of a KŁ function based on this property alone. The
answer is no and the reason is that a closed subspace of an ANR (absolute
neighborhood retract) is also an ANR if and only if it is a closed cofibration
(see [7, Proposition A.6.7]). The Alexander horned sphere, like any
topological manifold, is an ANR (see [10, Ch. III, Corollary 8.3]).

After these preliminary comments, we are ready to prove Theorem A. Let
$f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a KŁ (relative) function. For
the following result we will keep the notation of Proposition 3.6.

######  Proposition 4.1.

There exists a $C^{1}$ hypersurface $H\subset V\setminus f^{-1}(0)$ with the
following properties:

  1. (1)

The hypersurface $H$ intersects transversely every trajectory $\alpha$ of
$-\nabla f$ in $V$ exactly once.

  2. (2)

The restriction $R\bigr{|}_{H}\colon H\rightarrow\partial f^{-1}(0)$ of the
retraction $R\colon V\to f^{-1}(0)$ is a surjective, proper function.

###### Proof.

For item (1), we will write $f$ for the restriction $f\bigr{|}_{V}$, which
takes values in $[0,\rho)$. By the Implicit Function Theorem, the
hypersurfaces $f^{-1}(c)$ are $C^{1}$ for $c\in(0,\rho)$.

Define first an “abstract” $C^{1}$ manifold $\mathcal{H}$ of dimension $n-1$
as the inductive limit of the manifolds $f^{-1}(c)$ as $c\searrow 0$,
$c\in(0,\rho)$. Alternatively, we may describe $\mathcal{H}$ as follows.

If $c_{1}>c_{2}>0$, then the flow function gives an open $C^{1}$ embedding

$\displaystyle f^{-1}(c_{1})\hookrightarrow f^{-1}(c_{2})$
$\displaystyle\qquad x\mapsto\alpha_{x}\cap f^{-1}(c_{2}),$

where $\alpha_{x}$ is the trajectory determined by $x$.

On $V\setminus f^{-1}(0)$ consider the equivalence relation in which $x\sim y$
if the two points $x,y$ lie in the same trajectory. Take
$\mathcal{H}:=(V\setminus f^{-1}(0))/\sim$ with the induced quotient topology.
A set $U\subset\mathcal{H}$ is open if and only if $\widehat{U}:=\pi^{-1}(U)$
is a flow-invariant, open subset of $V\setminus f^{-1}(0)$, where

$\pi\colon V\setminus f^{-1}(0)\rightarrow\mathcal{H}$

is the canonical quotient map. Note that the topology is Hausdorff. Moreover,
since $f$ has no critical points in $V$, the flow is locally a product, which
implies that $\mathcal{H}$ is locally homeomorphic to $\mathbb{R}^{n-1}$. Now,
we need an atlas. For a fixed $c\in(0,\rho)$, we take the usual charts in
$f^{-1}(c)$ and project them via $\pi$ to $\mathcal{H}$. It is not hard to
verify that these charts are $C^{1}$-compatible, because the flow function is
$C^{1}$, thus concluding that $\mathcal{H}$ is a $C^{1}$ manifold. Moreover,
$\pi\colon V\setminus f^{-1}(0)\rightarrow\mathcal{H}$ is a $C^{1}$ function.

Let $h\colon\mathcal{H}\rightarrow[0,\infty)$ be an exhaustion function, i.e.,
a continuous, proper function. Let $\mathbb{N}/2:=\\{n/2\mid
n\in\mathbb{Z}_{\geq 0}\\}$ and, for every $k\in\mathbb{N}/2$, take the
covering of $\mathcal{H}$ by the relatively compact, open sets

$A_{2k+1}:=h^{-1}((k-1/3,k+1/3)).$

We let $\\{\phi_{2k+1}\\}_{k\in\mathbb{N}/2}$ be a $C^{1}$ partition of unity
subordinate to the covering $\\{A_{2k+1}\\}_{k\in}$ of $\mathcal{H}$. Note
that each point $x\in\mathcal{H}$ belongs to at most two of the sets
$A_{2k+1}$ and, consequently, at most two of the functions $\phi_{2k+1}$
satisfy $\phi_{2k+1}(x)\neq 0$. In fact, one can find a sufficiently small
neighborhood of $x$ that intersects at most two of the sets $A_{2k+1}$
nontrivially. By composing the $\phi_{2k+1}$ with $\pi$, we get $C^{1}$
functions

$\displaystyle\widehat{\phi}_{2k+1}\colon V\setminus
f^{-1}(0)\rightarrow[0,1],$
$\displaystyle\widehat{\phi}_{2k+1}:=\phi_{2k+1}\circ\pi.$

We will also consider the open sets $\widehat{A}_{2k+1}:=\pi^{-1}(A_{2k+1})$
in $V\setminus f^{-1}(0)$. For $n\in\mathbb{N}$, let

$\widehat{U}_{n}:=\bigcup_{2k+1\leq n}\widehat{A}_{2k+1}.$

Clearly, the sets $\widehat{U}_{n}$ are not relatively compact in
$f^{-1}(0,\rho)$.

We will need the following properties of compact subsets $K$ of $\mathcal{H}$,
which we will presently prove:

  1. (4.1)

There exists $c>0$ such that $K\subset\pi(f^{-1}(c))$. In particular, every
trajectory of $-\nabla f$ in $\pi^{-1}(K)$ intersects $f^{-1}(c)$.

  2. (4.2)

The intersection $\pi^{-1}(K)\cap f^{-1}(c)$ is a nonempty compact set in
$f^{-1}(c)$.

To prove (4.1), observe that the images of all preimages $f^{-1}(c)$ via $\pi$
are open subsets of $\mathcal{H}$. Their union covers $\mathcal{H}$ and, since
$K$ is compact, there will exist a $c$ such that $K\subset\pi(f^{-1}(c))$.
Indeed, first, compactness implies that $K$ is contained in the union of
finitely many sets $\pi(f^{-1}(a_{i}))$, with $a_{i}\in(0,\rho)$, $1\leq i\leq
k$ for some positive integer $k$. We can arrange for the $a_{i}$ to satisfy
$a_{i+1}<a_{i}$, which implies that
$\pi(f^{-1}(a_{i}))\subset\pi(f^{-1}(a_{i+1}))$. It follows that
$\bigcup_{i=1}^{k}\pi(f^{-1}(a_{i}))=\pi(f^{-1}(a_{k}))$. The second claim in
(4.1) is now an immediate consequence.

To verify (4.2), note that $\pi\bigr{|}_{f^{-1}(c)}$ is a homeomorphism onto
its image and

$\pi^{-1}(K)\cap f^{-1}(c)=(\pi\bigr{|}_{f^{-1}(c)})^{-1}(K).$

By properties (4.1) and (4.2), since the sets $U_{n}:=\bigcup_{2k+1\leq
n}A_{2k+1}$ are relatively compact in $\mathcal{H}$, there exists a decreasing
sequence $\rho>c_{1}>c_{2}>\ldots$ of positive numbers with $c_{n}\searrow 0$
and satisfying the following conditions:

  * •

The sets $\widehat{U}_{n}\cap f^{-1}(c_{n})$ are relatively compact in
$f^{-1}(c_{n})$.

  * •

Every trajectory of $-\nabla f$ in $\widehat{U}_{n}$ intersects
$f^{-1}(c_{n})$.

Define a positive function $\widehat{\Phi}$ on $V\setminus f^{-1}(0,\rho)$ by
letting

$\widehat{\Phi}:=\sum_{k\in\mathbb{N}/2}\frac{\widehat{\phi}_{2k+1}}{c_{2k+1}}.$

Recall that, by construction, at any given point in $V\setminus f^{-1}(0)$, at
most two of the functions $\widehat{\phi}_{2k+1}$ are nonzero. Thus, the
function

$\widehat{f}:=f\widehat{\Phi}$ (4.1)

is well-defined and $C^{1}$ on $V\setminus f^{-1}(0)$. Note that
$\widehat{\Phi}$ is constant on the trajectories of $-\nabla f$, which implies
that

$\langle\nabla\widehat{\Phi},\nabla f\rangle=d\widehat{\Phi}(\nabla f)=0.$

Hence

$\langle\nabla\widehat{f},\nabla f\rangle=\widehat{\Phi}|\nabla f|^{2}>0\mbox{
on }V\setminus f^{-1}(0).$

Therefore, $\widehat{f}$ is strictly decreasing along the trajectories of
$-\nabla f$. Moreover, $\widehat{f}$ can be continuously extended by $0$ along
$Z$.

Define $H:=\widehat{f}^{-1}(1)$. By what was just said, _if_ a trajectory of
$-\nabla f$ intersects $H$ it will do so only once and transversely.

Let $\alpha\colon(\mu^{-},\mu^{+})\rightarrow V\setminus f^{-1}(0)$ be a
trajectory of $-\nabla f$. Clearly,

$\lim_{t\rightarrow\mu^{+}}\widehat{f}(\alpha(t))=0.$

We will now show that

$\lim_{t\searrow\mu^{-}}\widehat{f}(\alpha(t))>1.$ (4.2)

From (4.2) and the fact that $\widehat{f}$ is strictly decreasing, it follows
that there exists $t_{0}\in\mathbb{R}$ such that
$\widehat{f}(\alpha(t_{0}))=1$ or, equivalently, that every trajectory
$\alpha$ intersects $H$.

In order to prove (4.2), let $k\in\mathbb{N}/2$ such that $p=\pi(\alpha(t))\in
A_{2i+1}\cap A_{2i+2}$. We have

$\displaystyle\widehat{f}(\alpha(t))=f(\alpha(t))\widehat{\Phi}(\alpha(t))=f(\alpha(t))\left(\frac{\phi_{2i+1}(\pi(\alpha(t)))}{c_{2i+1}}+\frac{\phi_{2i+2}(\pi(\alpha(t)))}{c_{2i+2}}\right).$
(4.3)

By the choice of the constants $c_{i}$ we know that the trajectory $\alpha$
which lies in $\widehat{U}_{2i+1}$ intersects $f^{-1}(c_{2i+1})$ at a time
$t^{\prime}$. Hence, for $t<t^{\prime}$ we have
$f(\alpha(t))>f(\alpha(t^{\prime}))=c_{2i+1}$. Therefore, for $t<t^{\prime}$
we have, by (4.3),

$\displaystyle\widehat{f}(\alpha(t))>c_{2i+1}\left(\frac{b(t)}{c_{2i+1}}+\frac{1-b(t)}{c_{2i+2}}\right)>c_{2i+1}\frac{b(t)+(1-b(t))}{c_{2i+1}}=1,$

where $b(t)\equiv\phi_{2i+1}(\pi(\alpha(t)))$. This proves (4.2) in the case
where $p\in A_{2i+1}\cap A_{2i+2}$. The case where $p\in
A_{2i+1}\setminus(A_{2i}\cup A_{2i+2})$ is dealt with in a similar fashion.
This finishes the proof of item (1).

We now prove item (2). Let us first show that $R\bigr{|}_{H}\colon
H\to\partial f^{-1}(0)$ is proper. Granted this, since a proper continuous
function to a locally compact Hausdorff space is closed, it will suffice to
prove that the image of $R\bigr{|}_{H}$ is dense to conclude that
$R\bigr{|}_{H}$ is also surjective.

We notice first that the continuous bijection
$\pi\bigr{|}_{H}:H\rightarrow\mathcal{H}$ is a homeomorphism. This can be seen
by noting that the flow function produces an open embedding
$f^{-1}(c)\hookrightarrow H$ for every $c\neq 0$ and, for every point
$p\in\mathcal{H}$, one finds an open neighborhood of $p\in\pi(f^{-1}(c))$.

Let $K\subset f^{-1}(0)$ be compact. By the previous paragraph, it will be
enough to prove that, for small $a>0$, the preimage $R^{-1}(K)\cap
f^{-1}(a)=(R\bigr{|}_{f^{-1}(a)})^{-1}(K)$ is compact in $f^{-1}(a)\subset V$.
Since the level sets of $f$ equal the level sets of $g$, we can reformulate
the last sentence with $g$ instead of $f$. Since $R^{-1}(K)\cap g^{-1}(a)$ is
closed, the only way it can fail to be compact is if there exists a sequence
$\\{x_{n}\\}\subset R^{-1}(K)\cap g^{-1}(a)$ such that
$x_{n}\rightarrow\partial g^{-1}(a)$. By this we mean that $x_{n}$ eventually
gets out of every compact subset of $g^{-1}(a)$. Then, clearly,

$\displaystyle\liminf d(x_{n},K)\geq d(K,\partial g^{-1}(a))\geq d(K,\partial
V)>0,$ (4.4)

where $\partial V:=\widehat{V}\setminus V$ is the boundary of $V$ within its
metric completion. The last inequality is a consequence of the local
compactness of $M$ and, therefore, of its open subset $V$. Indeed, one has
$d(x,\partial V)>0$ if $x\in V$, and it follows easily that $d(K,\partial
V)>0$ for every compact $K\subset V$.

Take now $a_{1}>0$ such that

${a_{1}}<{d(K,\partial V)}.$

By (3.9), the distance to $K$ from any point $p$ in $R^{-1}(K)\cap
g^{-1}(a_{1})$ (which, in particular, determines a trajectory with end-point
in $K$) will be at most $a_{1}$ and hence strictly smaller than $d(K,\partial
V)$. Therefore, (4.4) cannot be fulfilled for any sequence of points in
$R^{-1}(K)\cap g^{-1}(a_{1})$. Thus, we conclude that $R\bigr{|}_{H}$ is
proper.

Let us now prove that the image of $R\bigr{|}_{H}$ is dense in $f^{-1}(0)$. By
item (1), the image of $R\bigr{|}_{H}$ equals the image of $R$. Take a point
$p\in f^{-1}(0)$ and take an open ball around $p$. Then, by the uniform bound
on the lengths of trajectories of $-\nabla f$, there exists a smaller ball
around $p$ such that all the trajectories starting in the smaller ball cannot
leave the bigger ball. Hence the image of the limit function $R$ is dense and
this concludes the proof.

∎

The surjectivity of $R$ has the following trivial consequence.

######  Corollary 4.2.

If $p\in\partial Z$ is KŁ nondegenerate, then there exists a $C^{1}$ relative
curve $\gamma\colon([0,\epsilon),\\{0\\})\rightarrow(M,Z)$ such that
$|\gamma^{\prime}|\leq 1$ and $\gamma(0)=p$.

###### Proof.

Take $\gamma$ to be the parametrization by arc length of the integral curve of
$-\nabla f$ having $p$ as a limit point. ∎

We come now to the main result in this section, which finishes the proof of
Theorem A. Recall that the function $\widehat{f}$ was defined in (4.1) above.

######  Proposition 4.3.

The set $\widehat{f}^{-1}([0,1))$ is a $C^{1}$ regular open mapping cylinder
neighborhood of $f^{-1}(0)$.

###### Proof.

Recall that $Z:=f^{-1}(0)=\widehat{f}^{-1}(0)$. Let
$M_{R}:=H\times[0,1]\sqcup\partial Z/_{\sim}$ be the closed mapping cylinder
of $R\bigr{|}_{H}:H\rightarrow\partial Z$. Denote
$\partial^{[0,1]}\widehat{f}:=\widehat{f}^{-1}([0,1])\setminus\operatorname{Int}f^{-1}(0)=\widehat{f}^{-1}(0,1]\cup\partial
Z$.

The right-end function $q\rightarrow\omega_{q}^{+}$ for the initial value
problem for the vector field $-\frac{\nabla f}{|\nabla f|^{2}}$ is a
continuous function from $V\setminus f^{-1}(0)$ to $(0,\infty)$. In fact,
$\omega_{q}^{+}=f(q)$.

We can then fix a homeomorphism

$\displaystyle\upsilon\colon[0,1]\times H\rightarrow\\{(t,q)\mid q\in
H,\;t\in[0,\omega_{q}^{+}]\\}\subset\mathbb{R}\times H$ (4.5)

such that $\pi_{2}(\upsilon(t,q))=q$ and $\upsilon(0,q)=(0,q)$.

Define the function

$\displaystyle\Phi\colon M_{R}$
$\displaystyle\rightarrow\partial^{[0,1]}\hat{f}$ $\displaystyle\Phi(q,t)$
$\displaystyle=\begin{cases}\gamma_{q}(\upsilon(1-t,q)),&\text{ if
}t\in(0,1],\\\
\displaystyle\lim_{s\rightarrow\omega_{q}^{+}}\gamma_{q}(s),&\text{ if
}t=0,\end{cases}$

where $\gamma_{q}$ is the solution of the Cauchy problem for $-\nabla
f/|\nabla f|^{2}$ with initial value $q\in H$. Note that $\Phi$ is defined at
points $p=R(q)\in\partial Z$, in which case $\Phi(p)=\Phi(q,0)=R(q)=p$. We
also have that $\Phi\bigr{|}_{H\times\\{1\\}}=\operatorname{id}_{H}$.

Clearly, $\Phi\bigr{|}_{H\times(0,1]}$ is a continuous bijection onto
$\widehat{f}^{-1}(0,1]$, while $\Phi$ is the identity on $\partial Z$. Hence,
$\Phi$ is a bijection. It remains to prove that $\Phi$ and $\Phi^{-1}$ are
continuous.

Recall that, by item (2) in Proposition 4.1, the map $R\bigr{|}_{H}$ is
surjective. Then, we may identify the mapping cylinder $M_{R}$ with
$H\times[0,1]/_{\sim}$, where $(q_{1},0)\sim(q_{2},0)$ if and only if
$R(q_{1})=R(q_{2})$. Modulo this identification, and modulo the homeomorphism
$\upsilon$ defined in (4.5), the function $\Phi$ is the induced bijection of
the extension to $\\{(\omega_{q}^{+},q)~{}|~{}q\in H\\}$ of the flow function
of $-\frac{\nabla f}{|\nabla f|^{2}}$, a priori defined on

$D:=\\{(t,q)\mid q\in H,\;t\in[0,\omega_{q}^{+})\\}.$

More precisely, if $\phi\colon D\rightarrow\partial^{[0,1]}\hat{f}$ is the
flow function, then $\phi$ extends _continuously_ to a surjective function

$\widetilde{\phi}\colon\tilde{D}\rightarrow\partial^{[0,1]}\hat{f},$

where $\widetilde{D}:=\\{(t,q)\mid q\in H,\;t\in[0,\omega_{q}^{+}]\\}$. The
continuity of $\widetilde{\phi}$ is proved in the same manner as the
continuity of the retraction $R\colon V\rightarrow\partial Z$ in Proposition
3.6, while surjectivity is clear. By a standard result in point-set topology,
this induces a continuous bijection
$\tilde{D}/_{\sim^{\prime}}\rightarrow\partial^{[0,1]}\hat{f}$, where
$\sim^{\prime}$ identifies the points in the same fiber, i.e., _mutatis
mutandis_ , $\sim^{\prime}\equiv\sim$ in $H\times[0,1]/_{\sim}$. Adding to
this the fact that $\tilde{\phi}$ is proper, which follows along the same
lines as the properness of $R\bigr{|}_{H}$ in the proof of Proposition 4.1, we
conclude the continuity of the inverse, and hence of $\Phi^{-1}$. ∎

##  5\. Remarks on the KŁ nondegeneracy condition

The KŁ condition was introduced by Kurdyka in [12] in the o-minimal context,
where it served as the appropriate generalization of the Łojasiewicz
inequality. The characterization of the KŁ condition was thoroughly studied in
a nonsmooth, infinite dimensional setting in [2]. Theorems 18 and 20 in [2],
characterizing the KŁ property, are stated for proper, lower semicontinuos,
semi-convex functions on a Hilbert space. The context of relative manifolds
that we treat here is slightly different. In particular, in this section we do
not assume that $f$ is semi-convex, but rather ask for $f$ to be $C^{1}$ on
the regular part $M\setminus Z$. Therefore one may think of the results in
this section, in which we prove Theorem C, as complementing those in [2]. As a
matter of fact, one key lemma about upper semicontinuous functions from [2]
appears prominently here as well. We start by stating this result in a form
that applies to lower semicontinuous functions too.

######  Lemma 5.1 (cf. [2, Lemma 45]).

Let $u\colon(0,r_{0}]\rightarrow(0,\infty)$ be a locally integrable,
semicontinuous function. Then there exists a continuous function
$w\colon(0,r_{0}]\rightarrow(0,\infty)$ such that

  1. (1)

$w\leq u$, if $u$ is lower semicontinuous;

  2. (2)

$u\leq w$, if $u$ is upper semicontinuous; and

  3. (3)

$|u-w|\in L^{1}(0,r_{0})$.

######  Remark 5.2.

Note that the local integrability condition in the statement of the lemma is
automatically satisfied when $u$ is upper semicontinuous, in which case $u$ is
clearly locally bounded above.

###### Proof of Lemma 5.1.

We follow the proof of Lemma 45 in [2] and include the details when $u$ is
lower semicontinuous.

Let $\\{a_{k}\\}\subset(0,r_{0})$ be a strictly decreasing sequence with
$a_{k}\searrow 0$. We may justify the existence, for each $k$, of a continuous
function $w_{k}\colon[a_{k+1},a_{k}]\rightarrow(0,\infty)$ satisfying

$\displaystyle w_{k}\leq u$ (5.1)

and

$\displaystyle\int_{a_{k}}^{a_{k+1}}w_{k}\geq\int_{a_{k}}^{a_{k+1}}u-\frac{1}{(k+1)^{2}}$
(5.2)

using _Moreau envelopes_ $e_{\lambda}^{u}$ for lower semicontinuous functions
(see [22, Ch. 1.G]). These are given by

$e_{\lambda}^{u}(x)=\inf_{w}\left\\{u(w)+\frac{1}{2\lambda}|w-x|^{2}\right\\}\leq
u(x).$

The envelope $e_{\lambda}^{u}$ of a strictly positive lower semicontinuous
function $u$ defined on a compact set is a strictly positive continuous
function. When $\lambda\searrow 0$, one has $e_{\lambda}^{u}(x)\nearrow u$
pointwise. The integrability of $u$ on $[a_{k+1},a_{k}]$ implies that
$e_{\lambda}^{u}{{L^{1}\atop\xrightarrow{\hskip 11.38092pt}}\atop\;}u$, when
$\lambda\searrow 0$, so one can choose $\lambda$ such that (5.2) is satisfied
for $e_{\lambda}^{u}=:w_{k}$.

Let
$m_{k}:=\min\left\\{\frac{w_{k-1}(a_{k})}{w_{k}(a_{k})},\frac{w_{k}(a_{k})}{w_{k-1}(a_{k})}\right\\}$.
Clearly, $0<m_{k}\leq 1$. We construct a continuous function
$w\colon(0,r_{0}]\rightarrow(0,\infty)$ by induction on $k$ by letting

$w=\begin{cases}w_{k}&\text{on }[a_{k+1},a_{k}-\varepsilon_{k}],\\\
\lambda_{k}w_{k}&\text{on }[a_{k}-\varepsilon_{k},a_{k}],\end{cases}$

where $\lambda_{k}\colon[a_{k}-\varepsilon_{k},a_{k}]\rightarrow[m_{k},1]$ is
the unique affine function satisfying (necessarily)
$\lambda_{k}(a_{k}-\varepsilon_{k})=1$, $\lambda_{k}(a_{k})=m_{k}$, and where
$\varepsilon_{k}>0$ is to be chosen later. Then

$\displaystyle\int_{a_{k+1}}^{a_{k}-\varepsilon_{k}}w_{k}(r)~{}dr$
$\displaystyle=\int^{a_{k}}_{a_{k+1}}w_{k}(r)~{}dr-\int_{a_{k}-\varepsilon_{k}}^{a_{k}}w_{k}(r)~{}dr$
$\displaystyle\geq\int^{a_{k}}_{a_{k+1}}u(r)~{}dr-\frac{1}{(k+1)^{2}}-\int_{a_{k}-\varepsilon_{k}}^{a_{k}}w_{k}(r)~{}dr.$

Hence, for any choice of $\varepsilon_{k}>0$, we have

$\displaystyle\int^{a_{k}}_{a_{k+1}}w(r)~{}dr\geq\int^{a_{k}}_{a_{k+1}}u(r)~{}dr-\frac{1}{(k+1)^{2}}+\int_{a_{k}-\epsilon_{k}}^{a_{k}}(\lambda_{k}-1)w_{k}(r)~{}dr.$
(5.3)

Choose now $0<\varepsilon_{k}<a_{k}-a_{k+1}$ sufficiently small so that

$\displaystyle\int_{a_{k}-\varepsilon_{k}}^{a_{k}}(1-\lambda_{k}(r))w_{k}(r)~{}dr\leq\frac{1}{(k+1)^{2}}.$
(5.4)

Since, for any choice of $\varepsilon_{k}>0$, one has
$|(1-\lambda_{k}(r))w_{k}(r)|\leq\displaystyle\max_{[a_{k+1},a_{k}]}w_{k}$ for
all $r\in[a_{k}-\varepsilon_{k},a_{k}]$, we may choose an $\varepsilon_{k}$
satisfying (5.4). Hence, by inequality (5.3), we get

$\int^{a_{k}}_{a_{k+1}}w(r)~{}dr\geq\int^{a_{k}}_{a_{k+1}}u(r)~{}dr-\frac{2}{(k+1)^{2}}.$

Therefore, by induction,

$\int_{a_{k+1}}^{r_{0}}w(r)~{}dr\geq\int_{a_{k+1}}^{r_{0}}u(r)~{}dr-2\sum_{i=1}^{k+1}i^{-2}.$

Since, on every $[a_{k+1},a_{k}]$, we have $w\leq w_{k}\leq u$, we get that

$0\leq\int_{0}^{r_{0}}(u(r)-w(r))~{}dr\leq 2\sum_{{i=1}}^{{\infty}}i^{-2}.$

∎

With Lemma 5.1 and the following definition in hand, we are ready to prove
Theorem C, the second main result of this note.

######  Definition 5.3.

Let $(M,Z)$ be a relative manifold and let
$f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a relative $C^{1}$ function.

  1. (1)

A point $p\in\partial Z$ with a neighborhood $U_{p}$ such that
$\nabla_{x}f\neq 0$ for all $x\in U_{p}\setminus Z$ will be called _simple
nondegenerate_.

  2. (2)

A simple nondegenerate point $p\in\partial Z$ is weakly KŁ-nondegenerate if
there exists a triple $(\rho,U,\Psi)$ with $\rho>0$, $U\subset M$ an open
neighborhood of $p$, and
$\Psi\colon([0,\rho],\\{0\\})\rightarrow([0,\infty),\\{0\\})$ a relative
function which is nondecreasing and absolutely continuous on $[0,\rho]$, and
such that

$|\nabla_{x}(\Psi\circ f)|\geq 1\text{ for all }~{}x\in f^{-1}(D_{\Psi})\cap
U,$ (5.5)

where $D_{\Psi}\subset(0,\rho)$ are the points of differentiability of $\Psi$.

### Proof of Theorem C

For ease of reference, let us recall the statement of the theorem.

######  Theorem 5.4.

The following are equivalent for a simple nondegenerate point $p\in\partial
Z$:

  1. (1)

The point $p\in\partial Z$ is KŁ nondegenerate,

  2. (2)

The point $p\in\partial Z$ is weakly KŁ nondegenerate,

  3. (3)

There exists a compact neighborhood $K\ni p$ such that the upper
semicontinuous function

$\alpha^{K}(t)=\displaystyle{\frac{1}{\displaystyle\inf_{x\in f^{-1}(t)\cap
K_{p}}|\nabla_{x}f|}}$

is in $L^{1}(0,\rho)$ for some $\rho>0$.

  4. (4)

There exists an open neighborhood $U\ni p$, a positive number $\rho>0$, and a
continuous function $a\colon(0,\rho]\rightarrow(0,\infty)$ such that
$a^{-1}\in L^{1}(0,\rho)$, and

$\displaystyle|\nabla_{x}f|\geq a(f(x))\text{ for all }x\in U\setminus Z.$
(5.6)

###### Proof.

To show that $\eqref{T:KLchara_A}\Rightarrow\eqref{T:KLchara_B}$, it is enough
to prove that a continuous and increasing $\Psi$ which is $C^{1}$ on
$(0,\rho)$ is in fact absolutely continuous on $[0,\rho/2]$, since condition
(KŁ2) in Definition 2.1 will trivially imply condition (5.5). For this, it is
enough to check that the relation

$\Psi(t)-\Psi(s)=\int_{s}^{t}\Psi^{\prime}(r)~{}dr,$

valid a priori for $\rho/2\geq t\geq s>0$, also holds for $s=0$. Since
$\Psi^{\prime}\geq 0$, it follows that for any decreasing sequence
$s_{n}\searrow 0$ the sequence of functions
$g_{n}(r):=\xi_{[s_{n},t]}\Psi^{\prime}(r)$ is pointwise increasing and
nonnegative. Hence, by the Monotone Convergence Theorem,

$\int_{s_{n}}^{t}\Psi^{\prime}(r)~{}dr=\int_{[0,t]}g_{n}(r)~{}dr\longrightarrow\int_{0}^{t}\Psi^{\prime}(r)~{}dr.$

Due to the continuity of $\Psi$ in $0$, we get

$\Psi(t)-\Psi(0)=\Psi(t)-\lim_{n\rightarrow\infty}\Psi(s_{n})=\lim_{n\rightarrow\infty}\int_{s_{n}}^{t}\Psi^{\prime}(r)~{}dr=\int_{0}^{t}\Psi^{\prime}(r)~{}dr.$

By the Fundamental Theorem of Calculus, $\Psi$ is absolutely continuous on
$[0,\rho/2]$.

To prove that $\eqref{T:KLchara_B}\Rightarrow\eqref{T:KLchara_C}$, note first
that the upper semicontinuity of the function

$\alpha^{K}(t)=\sup_{x\in f^{-1}(t)\cap K}\frac{1}{|\nabla_{x}f|}$

for every compact set $K$ is straightforward. Since
$|\Psi^{\prime}|=\Psi^{\prime}$ on $D_{\Psi}$, one concludes that
$|\nabla_{x}(\Psi\circ f)|\geq 1$ implies that
$\Psi^{\prime}(f(x))\geq\frac{1}{|\nabla_{x}f|}$ for all $x\in
f^{-1}(D_{\Psi})\cap U$. Therefore,

$\Psi^{\prime}(t)\geq\displaystyle\sup_{x\in f^{-1}(t)\cap
U}\frac{1}{|\nabla_{x}f|}\text{ for all }t\in D_{\Psi}.$

Clearly, this implies that for every compact neighborhood $K\ni p$ with
$K\subset U$ we have

$\Psi^{\prime}(t)\geq\alpha^{K}(t).$

Integrating on $[0,\rho]$ and using the Fundamental Theorem of Calculus, we
get that $\alpha^{K}\in L^{1}(0,\rho).$

For $\eqref{T:KLchara_C}\Rightarrow\eqref{T:KLchara_D}$, we use Lemma 5.1 with
$u=\alpha^{K}$, for which we need that $\alpha^{K}$ is finite valued. Then
there exists continuous function
$\widetilde{a}\colon(0,\rho]\rightarrow(0,\infty)$ such that
$\widetilde{a}\geq\alpha^{K}$ and $\int_{0}^{\rho}\tilde{a}<\infty$. Let
$a:=1/\tilde{a}$. This means that

$|\nabla_{x}f|\geq\inf_{x\in f^{-1}(f(x))\cap
K}|\nabla_{x}f|=(\alpha^{K}(f(x)))^{-1}\geq a(f(x))\text{ for all }x\in K.$

To obtain the desired conclusion, we take $U$ to be the interior of $K$.

Finally, we show $\eqref{T:KLchara_D}\Rightarrow\eqref{T:KLchara_A}$. Define,
for $t\geq 0$,

$\Psi(t)=\int_{0}^{t}\frac{1}{a(r)}~{}dr.$

Clearly, $\Psi$ is increasing and $\Psi(0)=0$. For $t>0$, one has
$\Psi^{\prime}(t)=1/a(t)$, since we can write

$\Psi(t)=\int_{s}^{t}\frac{1}{a(r)}~{}dr+\int_{0}^{s}\frac{1}{a(r)}~{}dr$

for some fixed $0<s<t$. In other words, $\Psi\big{|}_{(0,\rho)}$ is $C^{1}$.
Then (5.6) can be written as

$|\nabla_{x}f|\geq\frac{1}{\Psi^{\prime}(f(x))},$

which is the same as $|\nabla_{x}(\Psi\circ f)|\geq 1$. ∎

######  Remark 5.5.

The equivalence of items (1) and (3) in Theorem 5.4 appears also in [2,
Theorems 18 and 20] when $f$ is assumed semi-convex, but only lower
semicontinuous.

######  Remark 5.6.

The continuity of the function $a$ in item (4) in Theorem 5.4 is not
essential. In fact, one can substitute it with the following condition.

  * $(4^{\prime})$

There exists an open neighborhood $U\ni p$, a positive number $\rho>0$, and a
Lebesgue measurable function $a\colon(0,\rho]\rightarrow(0,\infty)$ such that
$a^{-1}\in L^{1}(0,\rho)$ and

$\displaystyle\qquad|\nabla_{x}f|\geq a(f(x))\text{ for almost all }x\in
U\setminus Z$ (5.7)

Indeed, we have the implications
$\eqref{T:KLchara_D}\Rightarrow(4^{\prime})\Rightarrow(2^{\prime})\Rightarrow\eqref{T:KLchara_C}$,
where $(2^{\prime})$ is condition $\eqref{T:KLchara_B}$ in Theorem 5.4 with
(5.5) weakened to hold for almost all $x\in\;f^{-1}(D_{\Psi})\cap U$. By
examining the proof of Theorem 5.4, we get
$(2^{\prime})\Rightarrow\eqref{T:KLchara_C}$, while
$(4^{\prime})\Rightarrow(2^{\prime})$ follows the same path as
$\eqref{T:KLchara_D}\Rightarrow\eqref{T:KLchara_A}\Rightarrow\eqref{T:KLchara_B}$.
One has to be careful that, in (4), the condition that $p$ is simple
nondegenerate is redundant. This is not the case for $(4^{\prime})$ as stated,
which is why for Theorem 5.4 to hold one should keep simple nondegeneracy as a
“meta-condition”. Without it, $(4^{\prime})$ is what some authors call
_gradient inequality_ (see, for example, [11, Definition 2.2] and Remark 5.12
below).

######  Remark 5.7.

Note that condition (4) is trivially satisfied by a transnormal function
since, in that context (see [25, Lemma 1]), the integral $\int_{c}^{d}a^{-1}$
represents the distance between the parallel level sets $c$ and $d$ of $f$. In
this case the integral is finite because the distance is finite.

######  Remark 5.8.

We may apply Theorem 5.4 (4) when $U\setminus Z$ has a finite number of
connected components $U_{1},\ldots,U_{k}$ and, on each $U_{i}$, one can prove
the existence of a continuous function $a_{i}>0$ such that
$\int_{0}^{\rho}a_{i}^{-1}<\infty$ and (5.6) is satisfied on $U_{i}$. Take
$a:=\min\\{a_{1},\ldots,a_{k}\\}$ and note that one has $a^{-1}\in
L^{1}(0,\rho)$. Indeed, since $\sum_{i=1}^{k}a_{i}^{-1}\in L^{1}(0,\rho)$, the
claim follows from the fact that

$\displaystyle\frac{1}{a}<\sum_{i=1}^{k}\frac{1}{a_{i}}.$

Therefore, the function $a$ satisfies (5.6) on all of $U$. For example, when
$f^{-1}(c)$ is a critical level set of a Morse function $f\colon
M\rightarrow\mathbb{R}$, then $|f-c|$ is a KŁ function, since this is true for
the nonnegative functions $(f-c)^{+}:=\max\\{f-c,0\\}$ and
$(f-c)^{-}:=-\min\\{f-c,0\\}$ (see Example 2.2 (5)).

Here is a simple, if somewhat straightforward, application of what we have
done so far.

######  Corollary 5.9.

Let $v\in\mathbb{R}^{n}$ be a unit-norm vector and let
$M_{0}\subset\mathbb{R}^{n}$ be an oriented $C^{2}$ hypersurface with unit
normal $N_{x}$ at $x\in M_{0}$ contained in the open half-space

$H_{v}^{+}:=\\{x\in\mathbb{R}^{n}\mid\langle x,v\rangle>0\\}.$

Let $H_{v}:=\partial H_{v}^{+}$ and suppose that $p\in\overline{M_{0}}\cap
H_{v}$ is a point such that there exists a triple $(\rho,U,a)$, where
$\rho>0$, $U\ni p$ is a neighborhood, and
$a\colon(0,\rho)\rightarrow(0,\infty)$ is a continuous function with $1/a\in
L^{1}(0,\rho)$ and satisfying

$\displaystyle|v-\langle v,N_{x}\rangle|\geq a(\langle v,x\rangle)$

for all $x\in M_{0}\cap U\cap\\{x\mid\langle v,x\rangle\in(0,\rho)\\}$. Then
there exists a neighborhood $V\subset\overline{M}_{0}$ of $p$ and a continuous
function $\pi\colon V\rightarrow V\cap H_{v}$ such that $V$ is homeomorphic to
the mapping cylinder of $\pi$.

### Further observations

Theorem A, the main result of this article, shows, if anything, that the set
$Z$ cannot be too pathological in a neighborhood of a nondegenerate KŁ point.
The next result looks at the differential inequality that is, in a certain
sense, opposite to the one in item (4) of Theorem 5.4. By opposition to
Corollary 4.2 and the classical curve selection lemma in real analytic
geometry (see [15, Ch. 3]) one could say that the following result is a “no
curve selection lemma”.

######  Lemma 5.10.

Let $f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a $C^{1}$ relative
function. Suppose that there exists $\rho>0$ such that

$|\nabla_{x}f|\leq b(f(x))\text{ for all }x\in f^{-1}(0,\rho)$

for a continuous function $b\colon(0,\rho]\rightarrow(0,\infty)$ satisfying
$\int_{0}^{\rho}\frac{1}{b(t)}~{}dt=\infty$. Let $p\in\partial Z$. Then there
exists no relative $C^{1}$ curve
$\gamma\colon([0,\varepsilon),\\{0\\})\rightarrow(M,Z)$ with $\gamma(0)=p$ and
such that $|\gamma^{\prime}(t)|\leq 1$ for all $t\in(0,\varepsilon)$ fur
sufficiently small $\varepsilon>0$.

###### Proof.

We will proceed by contradiction. Suppose then that such a curve $\gamma$
exists. Let $\widetilde{\gamma}:=f\circ\gamma$ and note that
$\widetilde{\gamma}(0)=0.$ For all $t>0$,

$\displaystyle\widetilde{\gamma}^{\prime}(t)\leq|\widetilde{\gamma}^{\prime}(t)|=|\langle\nabla_{\gamma(t)}f,\gamma^{\prime}(t)\rangle|\leq|\nabla_{\gamma(t)}f|\leq
b(\widetilde{\gamma}(t)).$ (5.8)

Fix $t_{0}\in(0,\varepsilon)$ such that
$u_{0}:=\widetilde{\gamma}(t_{0})\in(0,\rho)$. Let
$B(u):=\displaystyle\int_{u_{0}}^{u}\frac{1}{b(r)}~{}dr$ and note that

$\displaystyle\lim_{u\searrow 0}B(u)=-\infty.$ (5.9)

From (5.8) we get, for $t<t_{0}$, that

$\displaystyle
B(\widetilde{\gamma}(t))=\int_{t_{0}}^{t}\frac{\tilde{\gamma}^{\prime}(s)}{b(\widetilde{\gamma}(s))}~{}ds\geq
t-t_{0}.$ (5.10)

Since $\displaystyle\lim_{t\searrow 0}\widetilde{\gamma}(t)=0$, inequality
(5.10) contradicts (5.9). ∎

Under the simple nondegeneracy condition, the functions which satisfy a
differential inequality as in Lemma 5.10 have the following alternative
description.

######  Proposition 5.11.

Let $(M,Z)$ be a relative manifold and let
$f\colon(M,Z)\rightarrow([0,\infty),\\{0\\})$ be a relative $C^{1}$ function.
Let $p\in\partial Z$ be a simple nondegenerate point. Then the following
assertions are equivalent:

  1. (1)

There exists $\rho>0$, an open neighborhood $U\ni p$, and a continuous
function $b\colon(0,\rho]\rightarrow(0,\infty)$ such that

$\int_{0}^{\rho}\frac{1}{b(r)}~{}dr=\infty$

and

$\displaystyle|\nabla_{x}f|\leq b(f(x))\text{ for all }x\in U\setminus Z.$
(5.11)

  2. (2)

There exists a compact neighborhood $K$ of $p$ such that the finite valued,
lower semicontinuous function
$\beta^{K}(t):=\displaystyle\frac{1}{\displaystyle\sup_{x\in f^{-1}(t)\cap
K}|\nabla_{x}f|}$ satisfies
$\displaystyle\int_{0}^{\rho}\beta^{K}(r)~{}dr=\infty$ for some $\rho>0$.

###### Proof.

For $\eqref{T:eqMZ_ITEM_A}\Rightarrow\eqref{T:eqMZ_ITEM_B}$, let $K\subset U$
be a compact neighborhood and note that condition (5.11) implies that
$1/\beta^{K}(t)\leq b(t)$. We get (2) by taking the reciprocals and
integrating.

To prove $\eqref{T:eqMZ_ITEM_B}\Rightarrow\eqref{T:eqMZ_ITEM_A}$, we use Lemma
5.1 to obtain a continuous function $\tilde{b}\colon(0,\rho]\to(0,\infty)$
with $0<\tilde{b}\leq\beta$ such that
$\displaystyle\int_{0}^{\rho}\tilde{b}=\infty$. Then
$b:=\tilde{b}^{-1}\geq(\beta^{K})^{-1}$ will satisfy item (1). ∎

It is unclear to us whether the situation contemplated in Proposition 5.11
really occurs under the hypothesis that $p\in\partial Z$ is a simple
nondegenerate point. One can easily construct examples of closed sets
$Z\subset\mathbb{R}^{2}$ for which there is no continuous curve in $M\setminus
Z$ with initial point $p\in\partial Z$.

This section suggests that the simple nondegenerate points $p\in\partial Z$,
where $Z$ is the zero locus of a $C^{1}$ function, may be divided in at most
three classes:

  * (i)

The “good” ones, i.e., those that are KŁ nondegenerate.

  * (ii)

The “bad” ones, i.e., those for which there exists a compact neighborhood $K$
such that

$\displaystyle\int_{0}^{\rho}\frac{1}{\sup_{x\in f^{-1}(t)\cap
K}|\nabla_{x}f|}~{}dt<\infty=\int_{0}^{\rho}\frac{1}{\inf_{x\in f^{-1}(t)\cap
K}|\nabla_{x}f|}~{}dt$

for some $\rho>0$. The examples in [2] and [3] which are not KŁ are of this
nature.

  * (iii)

The “ugly” ones, i.e., those for which there exists a compact neighborhood $K$
such that

$\displaystyle\int_{0}^{\rho}\frac{1}{\sup_{x\in f^{-1}(t)\cap
K}|\nabla_{x}f|}~{}dt=\infty.$

Examples in this class, if any, are still to be produced.

######  Remark 5.12.

If $n=1$ and $0\in\partial f^{-1}(0)$ is a simple nondegenerate point of a
$C^{1}$ relative function
$f\colon([0,\infty),\\{0\\})\rightarrow([0,\infty),\\{0\\})$, then $0$ is KŁ
nondegenerate. This follows by noting that $f$ is strictly increasing in a
neighborhood of $0$, hence bijective, and

$\inf_{x\in
f^{-1}\\{t\\}}\frac{1}{|f^{\prime}(x)|}=\frac{1}{f^{\prime}(f^{-1}(t))}=(f^{-1})^{\prime}(t),$

which is clearly in $L^{1}(0,\epsilon)$. Note that for $n=1$, condition
$(4^{\prime})$ without the simple nondegeneracy condition holds for every
$C^{1}$ function (see in [11, Theorem 3.2]).

## References

  * [1] James Waddell Alexander, _An example of a simply connected surface bounding a region which is not simply connected_ , Proc. Nat. Acad. Sc. 10 (1924), no. 1, 8–10.
  * [2] Jérôme Bolte, Aris Daniilidis, Olivier Ley, and Laurent Mazet, _Characterizations of lojasiewicz inequalities: subgradient flows, talweg, convexity_ , Trans. Amer. Math. Soc. 362 (2010), no. 6, 3319–3363. MR 2592958
  * [3] Jerome Bolte and Edouard Pauwels, _Curiosities and counterexamples in smooth convex optimization_ , hal-02447733v2 arXiv:2001.07999 (2020).
  * [4] C. E. Burgess and J. W. Cannon, _Embeddings of surfaces in $E^{3}$_, Rocky Mountain J. Math. 1 (1971), no. 2, 259–344. MR 278277
  * [5] Aris Daniilidis, Mounir Haddou, and Olivier Ley, _A convex function satisfying the lojasiewicz inequality but failing the gradient conjecture both at zero and infinity_ , hal-03136550 arXiv:2102.05342 (2021).
  * [6] James F. Davis and Paul Kirk, _Lecture notes in algebraic topology_ , Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR 1841974
  * [7] Rudolf Fritsch and Renzo A. Piccinini, _Cellular structures in topology_ , Cambridge Studies in Advanced Mathematics, vol. 19, Cambridge University Press, Cambridge, 1990. MR 1074175
  * [8] Piotr Hajłasz and Xiaodan Zhou, _Sobolev embedding of a sphere containing an arbitrary Cantor set in the image_ , Geom. Dedicata 184 (2016), 159–173. MR 3547787
  * [9] Morris W. Hirsch, Stephen Smale, and Robert L. Devaney, _Differential equations, dynamical systems, and an introduction to chaos_ , second ed., Pure and Applied Mathematics (Amsterdam), vol. 60, Elsevier/Academic Press, Amsterdam, 2004. MR 2144536
  * [10] Sze-tsen Hu, _Theory of retracts_ , Wayne State University Press, Detroit, 1965\. MR 0181977
  * [11] Sen-Zhong Huang, _Gradient inequalities_ , Mathematical Surveys and Monographs, vol. 126, American Mathematical Society, Providence, RI, 2006, With applications to asymptotic behavior and stability of gradient-like systems. MR 2226672
  * [12] Krzysztof Kurdyka, _On gradients of functions definable in o-minimal structures_ , Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 769–783. MR 1644089
  * [13] Kyung Whan Kwun and Frank Raymond, _Mapping cylinder neighborhoods_ , Michigan Math. J. 10 (1963), 353–357. MR 156324
  * [14] Stanisław Łojasiewicz, _Ensembles semi-analytiques,_preprint__ , IHES (1965).
  * [15] John Milnor, _Singular points of complex hypersurfaces_ , Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
  * [16] Edwin E. Moise, _Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung_, Ann. of Math. (2) 56 (1952), 96–114. MR 48805
  * [17] Victor Nicholson, _Mapping cylinder neighborhoods_ , Trans. Amer. Math. Soc. 143 (1969), 259–268. MR 248788
  * [18] Jacob Palis, Jr. and Welington de Melo, _Geometric theory of dynamical systems_ , Springer-Verlag, New York-Berlin, 1982, An introduction, Translated from the Portuguese by A. K. Manning. MR 669541
  * [19] Peter Petersen, _Riemannian geometry_ , third ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016. MR 3469435
  * [20] Frank Quinn, _Ends of maps. I_ , Ann. of Math. (2) 110 (1979), no. 2, 275–331. MR 549490
  * [21] by same author, _Lectures on controlled topology: mapping cylinder neighborhoods_ , Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), ICTP Lect. Notes, vol. 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, pp. 461–489. MR 1937021
  * [22] R. Tyrrell Rockafellar and Roger J.-B. Wets, _Variational analysis_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362
  * [23] Edwin H. Spanier, _Algebraic topology_ , Springer-Verlag, New York-Berlin, 1981, Corrected reprint. MR 666554
  * [24] Arne Strøm, _Note on cofibrations_ , Math. Scand. 19 (1966), 11–14. MR 211403
  * [25] Qi Ming Wang, _Isoparametric functions on Riemannian manifolds. I_ , Math. Ann. 277 (1987), no. 4, 639–646. MR 901710