arXiv research

A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…

Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.

New theorem on 3-manifolds with curvature and convex boundary.

The study proves splitting theorems for manifolds with specific curvature and boundary conditions.

Study of manifolds with specific curvature properties using capillary surfaces.

Develops a method to deform metrics on manifolds with non-compact boundaries.

Study rigidity of minimal disks in specific 3-manifolds.

The paper proves a theorem about splitting manifolds with specific curvature properties.

Sharp spectral extension of rigidity theorem for mean-convex manifolds.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…

Study proves rigidity of capillary surfaces in curved 3D spaces.

Study shows bound on Uryson width for specific 3D manifolds.

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

This article is based on a lecture at the Journal of Differential Geometry Conference, Harvard 2017. We discuss closed and torsion-free $G_{2}$-structures on a 7-manifold with boundary, with prescribed $3$-form on the boundary. Much of the article is based on an observation that there is an intrinsic notion of "mean co…

New Heintze-Karcher inequality helps understand droplet shapes.

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the bou…

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

The study bounds the topology of free boundary minimal surfaces in 3D manifolds.

Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.

The paper divides minimal hypersurfaces in a ball into two parts.

Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.

Proves existence of mean curvature flow with surgery for free boundary surfaces.

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $Σ$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that $Σ$ is locally volume-minimizing in a manif…

The paper classifies nilmanifolds with specific SL(3,C) structures.

Alternative solvability criterion for minimal surface equations and mean curvature flow.

The study proves inequalities for area and boundary length of disks in convex manifolds.

Given an unbounded domain $Ω$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the …

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.

Study resolves flow through cylindrical singularities, proving nonfattening.

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature vanishes on the boundary, then the metric must be flat.

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of…

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of prescribed boundary data. This result can be seen as a natural generalization of the…

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m…

The doubling conjecture for positive scalar curvature is proven under certain conditions.

Curvature conditions distinguish Euclidean space and disks in contractible manifolds.

In this paper, we consider compact free boundary constant mean curvature surfaces immersed in a mean convex body of the Euclidean space or in the unit sphere. We prove that the Morse index is bounded from below by a linear function of the genus and number of boundary components.

In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the bou…

Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a con…

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry. We also give an alternative proof that convex mean curvature flows with …

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient di…

Classifies 3-manifolds with uniformly positive scalar curvature.