arXiv research

New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.

Estimates the growth of Morse index for free boundary minimal hypersurfaces.

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

Study on unique minimal hypersurfaces in rotational domains.

Two minimal hypersurfaces in a ball intersect in any half-ball.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

The paper proves infinitely many free boundary minimal hypersurfaces in compact manifolds.

Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

The paper classifies stable free boundary minimal hypersurfaces outside a ball.

Study bounds index of minimal hypersurfaces in curved spaces.

Flat minimal hypersurfaces found in wedge-shaped domains.

Minimal surfaces' boundary points are always smooth.

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

Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.

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

Study finds new minimal surfaces in Schwarzschild space.

The paper generalizes free boundary min-max theory to equivariant settings.

Local minimality proven for stable free-boundary minimal hypersurfaces.

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 when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…

In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we characterize the orientation and Morse index of them.

New theorem finds new minimal hypersurfaces in hyperbolic space.

In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either $\frac{n^2}{4}$ or $\frac{(n-2)^2}{4|x|^2}$. Mor…

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates t…

Study min-max theory for hypersurfaces with boundary constraints.

Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.

Compact hypersurfaces minimize area in convex cones with free boundary.

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 generalization of the celebrated Schoen-Simon-Yau interior curvature estimates up to the free boundary. A direct corollary of our curvature estimates is…

The paper proves the existence of boundary minimal hypersurfaces in compact manifolds with boundary.

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $γ$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $γ$. In order to do so, we deve…

We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or e…

Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial Jacobi field. To approach this, we prove a one-sided Harnack inequality for min…

Proves intersection properties of minimal hypersurfaces in various spaces.

The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.

Let $(M^{n+1},\partial M,g)$ be a compact manifold with non-negative Ricci curvature, convex boundary and $2\leq n\leq 6$. We show that the min-max minimal hypersurface with respect to one-parameter families of hypersurfaces in $(M,\partial M)$ is orientable, of index one and multiplicity one.

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In part…

New mathematical surfaces without boundaries found.

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how …

The aim of this work is to study how the asymptotic boundary of a minimal hypersurface in H^nxR determines the behavior of the hypersurface at finite points, in several geometric situations.

Symmetric hypersurfaces and boundaries in R^n+1 with group actions.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence …

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic bou…

Develops theory for stable capillary minimal hypersurfaces in half-space.

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.