39 citations · 141 across the 13 of their papers we have counts for
14 papers
Boundary behaviour of stable Allen-Cahn limit-interfaces in Riemannian manifolds
Martin Man-chun Li, Davide Parise, Lorenzo Sarnataro
This is the second paper in our series of works to develop the min-max theory for free boundary minimal hypersurfaces in Riemannian manifolds with boundary via the Allen-Cahn equat…
Boundary behavior of limit-interfaces for the Allen-Cahn equation on Riemannian manifolds with Neumann boundary condition
Martin Li, Davide Parise, Lorenzo Sarnataro
We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen-Cahn energy functionals on a smooth bounded domain. Given any…
Contracting convex surfaces by mean curvature flow with free boundary on convex barriers
Sven Hirsch, Martin Li
We consider the mean curvature flow of compact convex surfaces in Euclidean -space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When th…
Min-max theory for free boundary minimal hypersurfaces II -- General Morse index bounds and applications
Qiang Guang, Martin Man-chun Li, Zhichao Wang +1
For any smooth Riemannian metric on an -dimensional compact manifold with boundary where , we establish general upper bounds for the Mors…
Free boundary minimal surfaces in the unit ball : recent advances and open questions
Martin Li
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few ye…
Free Boundary Minimal Surfaces in the Unit Three-Ball via Desingularization of the Critical Catenoid and the Equatorial Disk
Nikolaos Kapouleas, Martin Man-chun Li
We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical…
A maximum principle for free boundary minimal varieties of arbitrary codimension
Martin Li, Xin Zhou
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more gen…
Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann
Martin Li
We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a f…
Min-max theory for free boundary minimal hypersurfaces I - regularity theory
Martin Li, Xin Zhou
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s…
Curvature estimates for stable free boundary minimal hypersurfaces
Qiang Guang, Martin Li, Xin Zhou
In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces which satisfy a uniform area bound. Our result is a natural generalizati…
Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary
Ailana Fraser, Martin Li
We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact -dimensional manifold which has nonnegative Ricci curva…
A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary
Martin Li
In this paper, we develop a general existence theory for properly embedded minimal surfaces with free boundary in any compact Riemannian 3-manifold with boundary .…
A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary
Martin Li
Let be a compact -dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary . Assume that the mean curvature of the boundary…
On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature
Martin Li
We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold…