We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478v2] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.
We are concerned with the focusing $L^2$-critical nonlinear Schrödinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $t\to T$, $1\leq K<\infty$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of $K$ solitary waves with convergence rate $(1/t)^{2+}$, as $t\to \infty$. The uniqueness class is further enlarged to contain the multi-solitons with even lower …
This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps equation on 1+2 dimensional Minkowski space, with equivariance class k > 3. It is known that every topologically trivial wave map with energy less than twice that of the unique k-equivariant harmonic map Q scatters in both time directions. We study maps with precisely the threshold energy, i.e., twice the energy of Q. In the first part of the series we gave …
We study the equation \begin{equation*}\label{P0} (-Δ)^s u = |x|^α u^{\frac{N+2s+2α}{N-2s}}\mbox{ in }\mathbb{R}^N,\tag{P} \end{equation*} where $(-Δ)^s$ is the fractional Laplacian operator with $0 < s < 1$, $α>-2s$ and $N>2s$. We prove the linear non-degeneracy of positive radially symmetric solutions of the equation (\ref{P0}) and, as a consequence, a uniqueness result of those solutions with Morse index equal to one. In particular, the ground state solution is unique. Our non-degeneracy result extends in the radial setting some known theorems done by Dávila, Del Pino and Sire (see \cite[Th…
We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.
For the prescribed scalar curvature equation on $S^n$ ($n \ge 6$), we consider the situation where the number of bubbles tends to infinity in the Lyapunov-Schmidt (finite dimension) reduction method. In an outstanding paper by Wei and Yan, the special case where the bubbles are arranged "evenly" (close to a great circle) is considered. Here we are concerned with the generic scenario, where the bubbles are "planted" (arranged) in general position, in adjacent to the critical points of the prescribed function. The main interest of this article is to extract key information in the finite dimensio…
We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles $α\in (0,π)$. First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.
The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.
Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient conditions on a prescribed function $ϕ$ on $\mathbb{H}^3$, which ensure the existence of a ${\cal C}^1$-curve, parametrized by $\varepsilon\approx 0$, of embedded spheres in $\mathbb{H}^3$ having mean curvature $k +\varepsilonϕ$ at each point.
Here we provide a uniqueness result for viscosity solutions to sub-Riemannian mean curvature flow. In this setting the uniqueness cannot be deduced via comparison principle, which is known only for graphs and for radially symmetric surfaces, due to the presence of characteristic points. Here we prove that any viscosity solution is limit of a sequence of solutions of Riemannian flow, and obtain as a consequence uniqueness and comparison principle already known in the approximating riemannian setting. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and …
We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a "gradient flow calibr…
In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $Ω$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally sy…
In this paper we study resonances and eigenvalues for the nonlinear constant mean curvature eqn. linearized around the bubbles found by Brezis-Coron. This nonlinear eqn. is also called a H-system eqn. For degree one bubbles(the degree relates to a certain winding number) we only find resonances. For higher degree we prove eigenvalues do occur. Our goal is to eventually obtain dispersive estimates for the wave eqn. associated to the H-systems eqn. in its linear and non-linear form, a study of which was initiated by Chanillo-Yung.
Here we provide uniqueness of vanishing viscosity solutions to sub-Riemannian mean curvature flow problem, which was known only far from characteristic points or under special symmetry condition. We employ vanishing viscosity approach and look for solutions as limit of solutions to approximating flow, which is well defined also at characteristic points, and estimate the rate of convergence of the approximating solutions. The results are provided in the settings of both 3-dimensional rototranslation group $\text{SE}(2)$ and Heisenberg group $\text{H}^1$ and they are particularly important due t…
In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.
We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a strong uniqueness theorem when the cone is generic and use it to classify mean curvature flows coming out of generic cones of small entropy in low dimensions.
In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.
In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.
In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the mean curvature flow on $M^n \times [0,T]$ with bounded second fundamental forms. Suppose $F_T=\widetilde{F}_T$, then $F_t=\widetilde{F}_t$ on $M^n \times [0,T]$. This is an analog of a result of Kotschwar on the Ricci flow.
A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.
In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving m…
Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results if the mean curvature condition is replaced by an $σ_2$-curvature condition.
We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our st…
We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.
We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] -…
We show that any compact surface of genus zero in Euclidean 3-space that satisfies a quasiconformal inequality between its principal curvatures is a round sphere. This solves an old open problem by H. Hopf, and gives a spherical version of Simon's quasiconformal Bernstein theorem. The result generalizes, among others, Hopf's theorem for constant mean curvature spheres, the classification of round spheres as the only compact elliptic Weingarten surfaces of genus zero, and the uniqueness theorem for ovaloids by Han, Nadirashvili and Yuan. The proof relies on the Bers-Nirenberg representation of …
Stability and uniqueness of minimal disks in non-constant curvature
Universidad de la República（乌拉圭共和国大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；University of Chicago（芝加哥大学）；PUC-Rio（天主教里约热内卢大学）
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator $f$ is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is quantified in terms of t…
This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.
We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on the volume and the boundary area of K. We deduce this characterization from a new isoperimetric-type inequality for arbitrary convex bodies, for which the equality is achieved uniquely by balls. This second result is proved in a more general context of generalized mean-convex sets. Finally we positively answer a question left open in [FLW19] proving a furth…
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for…
