Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

If U:[0,+∞[×M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U:[0,+\infty [\times M$\end{document} is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂tU+H(x,∂xU)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$\end{document} where M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M$\end{document} is a not necessarily compact manifold, and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H$\end{document} is a Tonelli Hamiltonian, we prove the set Σ(U)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma (U)$\end{document}, of points in ]0,+∞[×M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$]0,+\infty [\times M$\end{document} where U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U$\end{document} is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(U)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Sigma (U)$\end{document}. We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.


Introduction
Let M be a smooth connected but not necessarily compact manifold. We will assume M endowed with a complete Riemannian metric g. For v ∈ T x M, the norm v x is g(v, v) 1/2 . We will denote also by · x the dual norm on T * x M. If γ : [a, b] → M is a curve, its length g (γ ) (for the metric g) is defined by The distance d that we will use on M is the Riemannian distance obtained from the Riemannian metric. Namely, if x, y ∈ M the distance d(x, y) is the infimum of the length of curves joining x to y. Since g is complete, the distance d is complete, and for every pair of points x, y ∈ M, there exists a curve joining x to y whose length is d(x, y). Moreover, for every compact subset K ⊂ M and every finite R ≥ 0, the closed R-neighborhood V R (K) = {x ∈ M | d(x, K) ≤ R} of K is itself compact.
Before giving our results for the general Hamilton-Jacobi equation, we will give the consequences in Riemannian geometry.
If C is a closed subset of the complete Riemannian manifold (M, g). As usual, the distance function d C : M → [0, +∞[ to C is defined by We will denote by * (d C ) the set of points in M \ C where d C is not differentiable. Note that * (d C ) has Lebesgue measure 0, since the Lipschitz function d C on M is differentiable almost everywhere. If U is a bounded connected component of M \ C, then U ∩ A * (C) = ∅, see §7, and Theorem 1.6 implies that the inclusion * (d C ) ∩ U ⊂ U is a homotopy equivalence. This fact was already known. It is due to Lieutier [21] in the Euclidean case and to Albano, Cannarsa, Nguyen & Sinestrari [1] in the general Riemannian case. The noncompact case is, to our knowledge, new, see however [7] where they study the unbounded components of * (d C ) in the Euclidean case.
A consequence of Theorem 1.6 is Of course, the homotopy equivalence in this theorem is a consequence of the compact version of Theorem 1.6, which is due, as we said above, to Albano, Cannarsa, Nguyen & Sinestrari [1].
We will give a version for non-compact M in §7.
For sake of completeness, we note that the subset M is a deformation retract of U (M, g). In fact, we can get such a retraction using the midpoint in a geodesic segment minimizing the length between the pair of points.
We now state our general results for Tonelli Hamiltonians. The local contractibility Theorem 1.8 is valid under slightly less restrictive conditions on the Hamiltonian H, see §8.
We recall that a Tonelli Hamiltonian H : T * M → R on M (for the complete Riemannian metric g) is a function H : T * M → R that satisfies the following conditions: (1*) The Hamiltonian H is at least C 2 (2*) (Uniform superlinearity) For every K ≥ 0, we have C * (K) = sup   Note that (2*) implies ∀(x, p) ∈ T * M, H(x, p) ≥ K p x − C * (K).
We will consider viscosity solutions of the Hamilton-Jacobi equation. There are several classical introductions to viscosity solutions [2,3,8,11]. The more recent introductions [14,15] are well-adapted to our manifold setting.
If u : N → R is a function defined on the manifold N, a singularity of u is a point of N where u is not differentiable. We denote by (u) the set of singularities of u.
The goal of this work is to study the topological structure of the set of singularities (U), with U : O → R a continuous viscosity solution of the evolutionary Hamilton-Jacobi equation defined on the open subset O ⊂ R × M.
In [9], we announced the results and sketched the proofs for M compact in the case of the stationary Hamilton-Jacobi equation, i.e. for U of the form U(t, x) = u(x) − ct, with u : M → R and c ∈ R. We extend the results of [9] to the case of the evolutionary Hamilton-Jacobi equation (1.1) covering also the case when M is non-compact.
Our first result is a local contractibility result. In fact, as we will see, the above theorem follows from its particular case with U : ]0, +∞[×M → R.
To give a more global result on the topology of (U) we need the Aubry set of a solution of (1.1). For this we first recall that the Lagrangian L : TM → R (associated to H) is defined by This Lagrangian L is finite everywhere, and enjoys the same properties as H, namely (1) The Lagrangian L is at least C 2 (in fact, it is as smooth as H).
Again (1.2) implies A Lagrangian L : TM → R, on the complete Riemannian manifold (M, g), is said to be Tonelli if it satisfies the conditions (1) to (4) above. It is well-known that U is differentiable at every point of I T (U), see Proposition 2.12. Therefore, we have (U) ∩ I T (U) = ∅. To avoid further machinery, in this introduction, we will state our results assuming the function U : [0, t] × M → R uniformly continuous.

Background
We will need to use some of the facts about viscosity solutions and the negative Lax-Oleinik semi-groups. We refer to [14] and [15] for details and proofs.
In the remainder of this work, we will assume that H : T * M → R is a given Tonelli Hamiltonian on the complete Riemannian manifold M. We will denote by L : TM → R its associated Lagrangian defined by

Action and minimizers
If γ : [a, b] → M is an absolutely continuous curve, its action L(γ ) is defined by Note that since L is bounded from below (by −C(0)), we always have L(γ ) > −∞, although we may have L(γ ) = +∞.
For x, y ∈ M, and t > 0, the minimal action h t (x, y) to join x to y in time t is where the infimum is taken over all absolutely continuous curves γ : From the definition of h t , we obtain the well-known inequality , for all t, s > 0 and x, y, z ∈ M.
A minimizer (for L) is an absolutely continuous curve γ : [a, b] → M such that Tonelli's theorem [6,10,13] states that, for every a < b ∈ R and every x, y ∈ M, there exists a minimizer γ : [a, b] → M, with γ (a) = x, γ (b) = y. All minimizers are as smooth as L.
Moreover, all minimizers are extremals, i.e. they satisfy the Euler-Lagrange equation given, in local coordinates, by As is well-known the 2nd order ODE (2.2) on M yields a 1st order ODE on TM which generates a flow φ L t on TM called the Euler-Lagrange flow. A curve γ : [a, b] → M is an extremal (i.e. satisfies (2.2)) if and only if its speed curve s → (γ (s),γ (s)) is (a piece of) an orbit of the Euler-Lagrange flow φ L t . An absolutely continuous curve γ : [a, b] → M is called a local minimizer (for L) if there exists a neighborhood U of γ ([a, b]) in M such that for any other absolutely continuous curve δ : [a, b] → U, with δ(a) = γ (a) and δ(b) = γ (b), we have L(δ) ≥ L(γ ). The regularity part of Tonelli's theorem implies that such a local minimizer γ : [a, b] → M is as smooth as L and satisfies the Euler-Lagrange equation (2.2). Therefore its speed curve s → (γ (s),γ (s)) is (a piece of) an orbit of the Euler-Lagrange flow φ L t .
is a curve, we will denote by L g (γ ) its action for this Lagrangian L g , that is The next lemma is well-known and follows from the Cauchy-Schwarz inequality.
with equality if and only if γ is a minimizing g-geodesic. Therefore, for every t > 0 and every x, y ∈ M, we have Moreover, a curve γ : [a, b] → M is L g -minimizing if and only if it is a minimizing geodesic.
The action of is given by Most of the proof of Lemma 2.4 is similar to or uses Lemma 2.2. In order to give the connection between action and viscosity solutions it is convenient to introduce the following (obvious) definition.
We now recall the definition of domination for a function.
Note that the right hand side of (2.6) makes always sense since we insisted that L(γ ) < +∞.
for every x, x ∈ M, and every t < t ∈ I.
A first connection between action and viscosity solutions is given by the following proposition, see [19] or [15] for a proof.  Again we used 2.9, rather than the more usual U(b, γ (b)) − U(a, γ (a)) = L(γ ), because we would like to allow possibly infinite values for U. Of course, the curve γ : [a, b] → M is said to be piecewise calibrated if we can find a finite sequence a = t 0 < t 1 The notion of calibrated curve is useful when U is a viscosity subsolution as can be seen from the following well-known proposition.
(3) the curve γ is a local minimizer. Hence it is as smooth as L and a solution of the Euler-Lagrange equation (2.2).
We will need some facts about differentiability and calibrated curves for viscosity subsolutions. We only quote the properties that we will use latter. The reader is referred to [2,3,8,[13][14][15][16] for context and proofs.
Moreover, the function U is differentiable at every point (t, γ (t)) with t ∈]a, b[.

The negative Lax-Oleinik semi-group and the negative Lax-Oleinik evolution
In fact, viscosity solutions which are continuous are always given by the negative Lax-Oleinik evolution as we now recall.
Once the minimal action is defined, we can introduce the negative Lax-Oleinik semi-group.
If u : M → [−∞, +∞] is a function and t > 0, the function T − t u : M → [−∞, +∞] is defined by We also set . This functionû is called the negative Lax-Oleinik evolution of u.
From the inequality (2.1), we obtain the well-known inequality 2t . Therefore, the associated negative Lax-Oleinik semi-group T g− t is defined, when t > 0, by If C ⊂ M, we define its (modified) characteristic function χ C : M → {0, +∞} by Therefore its negative Lax-Oleinik evolutionχ C , for the Lagrangian L g , is defined, for t > 0, by and d C (x) = inf c∈C d(c, x). Note that d C = dC, whereC is the closure of C in M. Hence, we getχ C =χC on ]0, +∞[×M. Therefore, to study the properties ofχ C on ]0, +∞[×M, we can always assume that C is a closed subset of M.
Proof. -Suppose that the curve γ : [a, b] → M, with a > 0, isχ C -calibrated. It must be a minimizing geodesic. We now recall, see Lemma 2.3, that for a minimizing g-geodesic γ : [a, b] → M, we have Therefore, using (2.11), we see that a minimizing g-geodesic γ : [a, b] → M, with a > 0, is calibrated if and only if This finishes to prove the first part with a > 0. The second part can be deduced from the first (using domination) and the fact that We will now give the relationship between the Aubry set I ∞ (χ C ) ofχ C , see Definition 1.9, and the Aubry set A * (C) of the closed set C, see Definition 1.4. Proof. -From Lemma 2.14, a curve γ : [0, +∞[→ M isχ C -calibrated for the Lagrangian L g , if and only if it is a minimizing g-geodesic satisfying Therefore, a constant curve with value in C isχ C -calibrating. This implies that Obviously, the curve γ : [0, +∞[→ M is a minimizing g-geodesic that satisfies (2.12), if and only γ λ is also a minimizing g-geodesic satisfying (2.12). Assume now that (t, y) ∈ I ∞ (χ C ), with y / ∈ C. We can find aχ C -calibrated curve γ : [0, +∞[→ M, with γ (t) = y. Since γ (0) ∈ C, the g-geodesic γ is not constant. Since geodesics are parametrized proportionally to arc-length, we can find λ such that γ λ is parameterized by arc length. As we saw above, the curve γ λ is a minimizing g-geodesic satisfying (2.12). Since the g-geodesic γ λ is minimizing and parametrized by arc-length, we get d C (γ λ (s)) = d(γ λ (s), γ λ (0)) = s. By Definition 1.4, this means that y ∈ A * (C).
The following lemma, left to the reader, sums up the calibration for the distance function to M .
The following theorems were obtained in [15]. This is a consequence of [15, Theorem 1.1].
This is a consequence of [ We now recall some more facts onû obtained, for example, in [15]. As a consequence of Proposition 2.20, without loss of generality, we can assume that the function u is lower semi-continuous when we consider properties ofû away from {0} × M. This is quite convenient since, as shown in [15] it allows to have backward characteristics and characterize the points whereû is differentiable.
Proposition 2.22 generalizes these facts, on the Lagrangian L g of Example 2.1, to all Tonelli Lagrangians. Therefore, recalling from Definition 1.2 that U (M, g) is the set of (x, y) ∈ M × M such that there exists a unique minimizing g-geodesic between x and y, we obtain that such that there exists at least two distinct minimizing g-geodesics between x and y-is the set

Cut points and cut time function
In this subsection, we will consider a lower semi-continuous Recall that t 0 (û) denotes the set of singularities ofû contained in ]0, t 0 [×M.
We now prove the density of the Sincê u is semiconcave and is differentiable everywhere on the open set V it is C 1,1 , see [8]. Therefore, if we set we conclude that the vector fieldX on V defined bȳ is locally Lipschitz. Therefore, we can find a unique solution : At this point, it is convenient to introduce the cut time function We give a characterization of the cut-time function.
Proof. -Property (i) is obvious. Property (ii) follows from Proposition 2.26. Property (iii) follows from Proposition 2.29 and the fact that calibrated curves are minimizers (hence extremals).
For part (iv) Assume that (t n , For n large, we have τ (t n , x n ) > t > t n . Therefore for such an n we can find aû-calibrated curve γ n : [0, t ] → M, with γ n (t n ) = x n . Extracting if necessary, we can obtain a curve γ : [0, t ] → M which is a C 1 limit of the minimizers γ n . This curve γ is alsoû-calibrated and satisfies γ (t) = x, this a contradiction since t > τ (t, x).

Local contractibility
Theorem 1.8 is a consequence of the following more general one.
Then the sets t 0 (û) and Cut t 0 (û) are locally contractible. In particular, they are locally path connected.
At this point it is useful to introduce the concept of U-adapted homotopy.
∈ t 0 (U), for some (t, x) ∈ S and some s ∈]0, δ], then the curve Of course, the important property of an adapted homotopy is the last one, as can be seen from the proof of the next Proposition.
Other nice features of U-adapted homotopies are the stability by restriction and composition, given in the following lemma whose proof is immediate.
is itself U-adapted.
We will deduce Theorem 3.1 from the lemma below, whose proof will be postponed to section §4. Then, for every compact subset C ⊂]0, t 0 [×M, we can find δ > 0 and aû-adapted homotopy F : C × [0, δ] → M such that for every (t, x) ∈ C and every s ∈]0, δ], we have Remark 3.7. -In the proof of Theorem 3.1, we only use the existence of aûadapted homotopy. The inequality (3.1) will be used for the application to Riemannian manifolds. More precisely, we will use inequality (3.1) to obtain Proposition 7.3, which is used in the proof of Theorem 1.6. and a neighborhood of (t,x) We now remark that, to prove the theorem, it suffices to find a neighborhood [t − ,t + ] × V of (t,x) contained in [a, b] × K and a homotopy H : In fact, properties (C1) and (C2) show that the inclusion . We now observe that, cutting down the neighborhood V ofx on the manifold M, we can assume that V is contractible in itself. Hence, so is [t − ,t + ] × V. Therefore, by (C3), we obtain that H(·, 1) on [t − ,t + ] × V is homotopic to a constant as maps with values in t 0 (û) ∩ [a, b] × K. This clearly finishes the proof of local contractibility for both t 0 (û) and Cut t 0 (û).
It remains to construct H. We first use Lemma 3.6 to find aû-adapted homotopy F : where τ is the cut time function ofû. Using that τ is upper semi-continuous, we conclude that τ (t, x) − t < δ in a neighborhood of (t,x) in ]0, t 0 [×M. Hence, cutting down and V if necessary, we can assume that τ (t, x)−t < δ, for (t, x) ∈ [t − ,t + ]×V. Therefore, Proposition (3.4) applied to theû-adapted homotopy F implies We now define the continuous homotopy H : Obviously, the homotopy H satisfies the required property (C1) given above. It also We now prove Theorem 1.1.
Proof of Theorem 1.1. -If C is a closed subset of the complete Riemannian manifold M, from Example 2.13, the negative Lax-Oleinik evolutionχ C is given, for t > 0, byχ The partial derivative ∂ tχC is given by Hence, it is defined and continuous everywhere. This implies that where (d 2 C ), as usual, is the set of points in M where d 2 C is not differentiable. From Theorem 3.1, we obtain that (χ C ) =]0, +∞[× (d 2 C ) is locally contractible, which implies that (d 2 C ) is also locally contractible. We now observe that d 2 C is differentiable at every point c ∈ C, since 0 ≤ d 2 This finishes the proof Theorem 1.1.
Proof. -We first show the second part of this Lemma. Since we will prove the uniqueness by the end of the proof, for now, we will show that for a given r > 0, the maximum max z∈Mû (t + s, z) − h s (x, z), for s small, can only be achieved inB(x, r).
In particularũ(t, s, x) is finite when t + s < t 0 and continuous at every (t, 0, x), with t < t 0 .
For the right hand side, we haveũ(t, s, x) = max z∈Mû (t + s, z) − h s (x, z) ≥û(t + s, x) − h s (x, x). But h s (x, x) is less than the action of the constant path at x, hence
Since y ∈V 2 (K) is compact, the equality 4.5 (together with Claim 4.2 to cover the case s = 0) implies that the functionũ is continuous on To prove the uniqueness part of Lemma 4.1, we need to introduce the positive Lax-Oleinik semi-group T + t , t ≥ 0, whose definition we now recall. We also set T + 0 u = u.  We will callǔ the positive Lax-Oleinik evolution.
Recalling thatû(t + s, y) = T − t+s u(y), for t, s, we see that This equality is also valid for s = 0, from the definition of T + 0 andũ.
The main point to prove the uniqueness statement in Lemma 4.1 is the following Lemma. Proof. -As we know already from Claim 4.6 that T + s T − s+t u is locally semi-convex, it suffices to prove that T + s T − s+t u is locally semi-concave on a neighborhood ofV 1/2 (K). Sinceû is locally semi-concave on ]0, t 0 [×M, we obtain that the family of functions (T − t u) t∈[a− ,b+ ] is equicontinuous and equi-semiconcave on the compact neighborhood If M is compact, we could take K = M, and the C 1 property above follows from [5].
For the noncompact case, we will use [17,Appendix B], which adapts some of the results of [5] to the noncompact setting.
As we will show below, the construction of the homotopy in Lemma 3.6 will follow easily from the uniqueness given in Lemma 4.1.  h s (x, y). Moreover, if s ≤ min(δ, δ(r)), where δ(r) is given by Lemma 4.1, we have d(y, x) ≤ r, for the y given by (4.8).
Therefore, the existence of y = y(t, s, x), as we saw, follows from (4.5).
We now show the uniqueness. Suppose that y = y(t, s, x) is a point where u(t + s, y) − h s (x, y) achieves its maximumũ(t, s, x). Calling γ : [0, s] → M the minimizer such that γ (0) = x and γ (s) = y, we have γ (s)). Since T + s T − s+t u is differentiable at x = γ (0), we use the T + t version of Proposition 2.12 (again this version follows from Remark 4.5) to obtain that  Proof. -We first show that F is continuous on ([a, b] × K)×]0, δ]. For this, we remark that, since δ ≤ δ(1), the map F takes values in the compact setV 2 (K). Therefore, to show that F is continuous on ([a, b] × K)×]0, δ], it suffices to show that its graph is closed in ([a, b] × K)×]0, δ] ×V 2 (K). By the definition of F, its graph Graph(F) is given by Since, as we observed above, the three functions (t, s, x) →ũ(t, s, x), (t, s, y) → T − s+t u(y) and (s, x, y) → h s (x, y) are all continuous for (t, s, x, y) ∈ [a, b] × ]0, δ] × K × M, we conclude that Graph(F) is closed.
To show that F is continuous at a point in We now check condition (3) by the T + τ version of Proposition 2.12, we obtain We now observe that, by the same Proposition 2.12, the backwardû-characteristicγ : [0, t + s] → M ending at γ (t + s) satisfies Since γ (t + s) =γ (t + s), we conclude that the two extremals γ andγ have the same position and speed at time t + s, therefore γ =γ |[t, t + s]. In particular, the curve γ iŝ u-calibrated.
To finish the proof of Lemma 3.6, it remains to observe that (3.1) follows from the definition of F, and that any compact subset  Note also that, if the adapted homotopies F 1 , F 2 of Lemma 3.5 satisfy condition (CC), so does the homotopy F obtained from them in that Lemma 3.5.
Therefore, as the reader, will realise, all adapted homotopies that we construct will also satisfy condition (CC).
It is not difficult to check that H satisfies the required properties.
This finishes the proof of Proposition 5.1.
Here is a useful criterion that allows to show that the hypothesis of Proposition 5.1 holds. We first find a, b ∈]0, +∞[, with a < b, and a compact subset K ⊂ M such that C ⊂ [a, b] × K.
For n = 1, we just take asF the restriction of F to C × [0, δ].
Since n 0 δ > 1, this finishes the proof of the Lemma.

Functions Lipschitz in the large
To state the generalization we have in mind, we recall the definition of Lipschitz in the large for a function, see [22,Definition A.5] or [15]. Definition 6.1. -Let X be a metric space whose distance is denoted by d. A function u : X → R is said to be Lipschitz in the large if there exists a constant K < +∞ such that |u(y) − u(x)| ≤ K + Kd(x, y), for every x, y ∈ X.
When the inequality above is satisfied, we will say that u is Lipschitz in the large with constant K.
Note that we do not assume in the definition above that u is continuous. Obviously, when X is compact u : X → R is Lipschitz in the large if and only if u is bounded.
As is shown in [15,Proposition 10.3], the function u : X → R is Lipschitz in the large if and only if there exits a (globally) Lipschitz function ϕ : X → R such that In particular, a Lipschitz in the large function u : M → R is bounded from below by a Lipschitz function and thereforeû is finite everywhere on [0, +∞[×M.
As we will see below the next theorem generalizes Theorem 1.10 stated in the introduction. It is not difficult to see that, for a function u : M → R Lipschitz in the large with constant K, its lower semi-continuous regularization u − is itself Lipschitz in the large with constant K. Therefore, by Proposition 2.20, without loss of generality we can prove Theorem 6.2 adding the assumption that u is lower semi-continuous.
To prove Theorem 6.2, we need a preliminary result, namely Corollary 6.4 below. To obtain this Corollary, we need following result, whose proof is standard and can be found in [15,Theorem 10.4].
Proof. -By Proposition 6.3 above we can find a global Lipschitz constant λ < +∞ forû on But by the uniform superlinearity of L, see (1.4) U(0, x). Note that the function u itself is uniformly continuous.
Any uniformly continuous function u : M → R is Lipschitz in the large, since it is a uniform limit of Lipschitz functions, see for example [15,Lemma A.1]. Therefore, Theorem 1.10 is a consequence of the more general Theorem 6.2. By compactness of C, there exists t 0 > 0 such that C ⊂]t 0 , +∞[×M.
x) ∈ C} ⊂ ]0, +∞[, since u t 0 is Lipschitz in the large, by the beginning of the proof of Theorem 6.2, we know that we can find aû t 0 -adapted homotopy F 0 : t 0 , x), is aû-adapted homotopy. Proof. -By Theorem 2.18, we know that T − t u is continuous on M, for every t > 0, therefore Lipschitz in the large, since M is compact. It suffices now to apply Corollary 6.5. In particular, if M is compact, then for any closed subset C we have I +∞ (χ C ) =]0, +∞[×C.
For the second part, we note that, for a non-constantχ C -calibrated curve γ :

More applications to complete non-compact Riemannian manifold
Assume that C is a closed subset of the complete Riemannian manifold (M, g). If M is not compact, then neither χ C norχ C (t 0 , ·) = d C (·) 2 /2t are necessarily Lipschitz in the large. So, we cannot apply Corollary 6.5 to obtain the global homotopy type of * (d C ) = (d 2 C ). Instead, we will show that Proposition 5.1 directly applies. For y ∈ M and t, s > 0, we introduce the function ϕ t,s,y : M → R defined by Proof. -From the definition (7.1), the inequality ϕ t,s,y (x) ≥ ϕ t,s,y (y) translates to from which the inequality d(x, y) ≤ 2sd C (y)/t follows. To prove the last inequality of the lemma, we again use x 0 ), . . . , (t n , x n ), . . . is a (finite or infinite) sequence in ]0, +∞[×M, with s n = t n+1 − t n ≥ 0, and such that ϕ t n ,s n ,x n (x n+1 ) ≥ ϕ t n ,s n ,x n (x n ). Then Proof. -From Lemma 7.1, we get (7.2) d(x n+1 , x n ) ≤ 2s n t n d C (x n ) and d C (x n+1 ) ≤ 1 + 2s n t n d C (x n ).
from the second inequality above. The first inequality in the lemma follows from n−1 i=0 1 + 2s i t i ≤ e 2t n /t 0 , which we now establish by taking log's. Using that log(1 + t) ≤ t, for t ∈]0, +∞[, that t i is non-decreasing and n−1 We now combine the already established first inequality of the lemma with the inequality (7.2) to obtain Combining this last inequality with t i non-decreasing, we obtain Therefore, since n−1 i=0 s i ≤ t n and t ≤ exp(t), we get We now prove that χ C satisfies the hypothesis of Proposition 5.1. Since A is compact, its neighborhoodV κ (A) = {x ∈ M | d A (x) ≤ κ} is also compact. By Lemma 3.6, we can find δ > 0 and aχ C -adapted homotopy F : [a, b + 1] ×V κ (A) × [0, δ] → M.
If δ ≥ 1, then the restriction of F to K × [0, 1] does the job. If not, cutting down on δ we assume δ = 1/n, with n an integer ≥ 2.
This homotopy is well-defined if we show by induction that, for (t, x) ∈ K ⊂ [a, b] × A, the sequence (x i , t + i/n), with x 0 = x, defined by induction for i = 1, . . . n − 1 as is such that x i ∈V κ (A), for i = 1, . . . , n − 1.
We first introduce the subset AU (M, g) ⊂ M × M.  The proof is left to the reader. The following generalization of Theorem 1.7 now follows from the lemma above and Theorem 1.6.

More results on local contractibility
In fact, our local contractibility result can be applied for viscosity solution defined only on an open subset, and also for Hamiltonians which are not necessarily uniformly superlinear. More precisely, we have.  (x, p). Since the support of ϕ is compact, using the properties of H, it is not difficult to check that H is Tonelli and coincides with H on T * W. This last fact implies that U : ]a, b[×W → R is also a viscosity solution of (1.1) forH.