arXiv subjects

Carlo Sinestrari

Publications and source records attributed to Carlo Sinestrari.

17 recordsLinked to original sources

arXiv2026

Obstacles and Singularities of Riemannian Distance Functions

We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton--Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.

math.AP↗

arXiv2025

Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat $3$-manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and the initial surface is round in a suitable sense, then the flow exists for all times and converges smoothly to a stable CMC surface. This extends to the asymptotically flat setting a classical result by Huisken-Yau (Invent. Math. 1996) and allows to construct a CMC foliation of the outer part of the manifold by an alternative approach to the ones by Nerz (Calc. Var. PDE, 2015) or by Eichmair-Koerber (J. Diff. Geometry, 2024).

math.DG↗

arXiv2024

Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as $t \to +\infty$

math.DG↗

arXiv2022

Non-homothetic convex ancient solutions for flows by high powers of curvature

We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree greater than one. This generalises previous work on the mean curvature flow and other one-homogeneous curvature flows. As an auxiliary result, we prove a new theorem on the convergence to a round point of convex rotationally symmetric hypersurfaces satisfying a suitable constraint on the curvatures.

math.DG↗

arXiv2020

Strong spherical rigidity of ancient solutions of expansive curvature flows

We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is proved using a reflection argument. Our theorem shows that expanding flows exhibit stronger spherical rigidity, if compared with the classification results of ancient solutions in the contractive case.

math.DG↗

arXiv2019

Generation of singularities from the initial datum for Hamilton-Jacobi equations

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point.

math.AP↗

arXiv2018

Convex sets evolving by volume preserving fractional mean curvature flows

We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics approach round spheres. The proofs are based on apriori estimates on the inner and outer radii of the solutions.

math.AP↗

arXiv2018

Ancient solutions of geometric flows with curvature pinching

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.

math.DG↗

arXiv2018

Volume preserving flow by powers of symmetric polynomials in the principal curvatures

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.

math.DG↗

arXiv2016

Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the monotonicity of the isoperimetric ratio, which allows to control the inner and outer radius of the hypersurface and to deduce uniform bounds on the curvature by maximum principle arguments.

math.DG↗

arXiv2016

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.

math.DG↗

arXiv2016

Neckpinch singularities in fractional mean curvature flows

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when $n = 2$, our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.

math.DG↗

arXiv2016

Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. The latter behavior is only possible if the dimension is even. These results generalize previous works by Huisken and Baker on the mean curvature flow of submanifolds of the sphere.

math.DG↗

arXiv2014

Global Propagation of Singularities for Time Dependent Hamilton-Jacobi Equations

We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the form \begin{equation*} u_t(t,x)+H(\nabla u(t,x))=0, \qquad\text{a.e. }(t,x)\in (0,+\infty)\timesΩ\subset\mathbb{R}^{n+1}\,. \end{equation*} It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian $H$. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of the above equation cannot vanish in a finite time.

math.AP↗

arXiv2013

Lagrangian mean curvature flow of pinched submanifolds of CP^n

We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a Lagrangian submanifold satisfying our pinching condition is diffeomorphic to a real projective space.

math.DG↗

arXiv2011

Singular gradient flow of the distance function and homotopy equivalence

It is a generally shared opinion that significant information about the topology of a bounded domain $Ω$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partialΩ}$, %, $d:Ω\rightarrow [0,\infty [$, from the boundary of $Ω$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partialΩ}$. As an application, we deduce that such a singular set has the same homotopy type as $Ω$.

math.AP↗

arXiv2009

Volume-preserving flow by powers of the m-th mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.

math.DG↗