arXiv research

Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show …

The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.

The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.

Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.

Generic scarring occurs along stable minimal hypersurfaces in 3-7 dimensional manifolds.

We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar end…

The study proves that certain minimal hypersurfaces in 4D space must be planes.

Lower bound found for eigenvalue of hypersurface in Riemannian manifold.

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…

A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.

Proves intersection properties of minimal hypersurfaces in various spaces.

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…

New tensors capture intrinsic embedding data of conformal hypersurfaces.

The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.

Minimal hypersurfaces scarring along a fixed one in certain manifolds.

The paper constructs stable minimal hypersurfaces with specific singularities.

We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.

Localized min-max method proves minimal hypersurface existence.

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

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

Ancient mean curvature flows start from unstable minimal hypersurfaces.

Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated…

Paper confirms Yau's conjecture about sphere eigenvalues.

New minimal hypersurfaces in 4D sphere found.

Flat minimal hypersurfaces found in wedge-shaped domains.

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…

A minimal hypersurface in a sphere is uniquely determined.

Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold ($2\le n\le 6$) that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min-max methods: they have index at most $1$. We apply this to obtain a lower area bound for su…

For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.

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

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

Improved estimate for eigenvalues of minimal hypersurfaces in spheres.

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic …

In 3-dimensional Euclidean space, Scherk second surfaces are singly periodic embedded minimal surfaces with four planar ends. In this paper, we obtain a natural generalization of these minimal surfaces in any higher dimensional Euclidean space ${\R}^{n+1}$, for $n \geq 3$. More precisely, we show that there exist $(n-1…

Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.

Infinite $G$-invariant minimal hypersurfaces found in Riemannian manifolds.

The study characterizes embedded minimal hypersurfaces in $S^{n+1}$ with symmetries.

Study proves higher-order conformal forms don't exist in odd dimensions.

We show that a bumpy closed Riemannian manifold $(M^{n+1}, g)$ $(3 \leq n+1 \leq 7)$ admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a previous result by O. Chodosh and C. Mantoulidis on connected minimal hypersurfaces wit…

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2\leq n\leq 6$. We characterize the Morse index, area and multiplicity of this min-max hyp…

Minimal hypersurfaces in spheres generated by isoparametric foliations are found.

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

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of …

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…

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the e…

Study finds new minimal surfaces in Schwarzschild space.