arXiv research

Sharp geometric inequalities for free boundary hypersurfaces in balls.

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

Optimal inequality for free boundary hypersurfaces in convex domains.

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

Study on unique minimal hypersurfaces in rotational domains.

The paper classifies CMC free boundary hypersurfaces in rotational domains.

Paper generalizes Minkowski inequality for umbilical hypersurfaces with free boundary.

The paper explores geometric properties of free boundary hypersurfaces in balls.

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

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

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

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

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

Study finds rigid properties of boundary-free hypersurfaces in specific data sets.

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

Proves rigidity of stable free boundary hypersurfaces in 5-manifolds.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

Study curves evolving on hypersurfaces with free boundaries, preserving length.

Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.

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

Compact hypersurfaces minimize area in convex cones with free boundary.

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, pro…

Flat minimal hypersurfaces found in wedge-shaped domains.

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

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

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…

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.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkow…

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…

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

Local minimality proven for stable free-boundary minimal hypersurfaces.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

Study bounds index of minimal hypersurfaces in curved spaces.

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…

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…

Study finds new minimal surfaces in Schwarzschild space.

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension $n\geq 3$ with boundary on the sphere.

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…

We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…

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 $V$.

New mathematical surfaces without boundaries found.

The paper examines stable capillary hypersurfaces in hyperbolic space.

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…

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

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

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

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 …

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…