arXiv research

New tensors capture intrinsic embedding data of conformal hypersurfaces.

New compact mean convex hypersurfaces found for positive λ.

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

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

Paper constructs a new type of hypersurface in Euclidean spaces.

New types of Delaunay hypersurfaces found in spheres.

New $λ$-hypersurfaces not isometric to standard spheres.

Estimates for spacelike hypersurfaces in de Sitter space.

The paper constructs $λ$-hypersurfaces for $λ>0$ and $λ<0$.

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 study characterizes geometries of hypersurfaces in warped product and conformal manifolds.

This paper proves the manifold hypothesis for lower embedding dimensions using osculating hyperspheres.

Lower bound found for eigenvalue of hypersurface in Riemannian manifold.

The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.

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

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

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…

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

Estimates the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.

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…

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

Study on hypersurfaces in Einstein manifolds using Killing spinors.

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…

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m>0$ in $S^{n+1}(1)$, for $1\leq m\leq n-1$. In particular, if the $4^{\text{th}}$…

The paper finds new constant mean curvature hypersurfaces in spheres.

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

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…

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

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\R^{n}$ (Delaunay unduloids). When $n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfac…

Revisits and applies a formula for hypersurface Euler characteristics.

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

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

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

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfac…

The paper constructs stable minimal hypersurfaces with specific singularities.

We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first resu…

Proves intersection properties of minimal hypersurfaces in various spaces.

In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r…

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…

Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures

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.

In this paper we show the existence of a closed, embedded $λ$-hypersurfaces $Σ\subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n) \times SO(n)$ symmetry. Our approach uses a "shooting method" similar to the approach used by McG…

We prove a non-collapsing property for curvature flows of embedded hypersurfaces in the sphere and in hyperbolic space.

We consider the smooth inverse mean curvature flow of strictly convex hypersurfaces with boundary embedded in $\mathbb{R}^{n+1},$ which are perpendicular to the unit sphere from the inside. We prove that the flow hypersurfaces converge to the embedding of a flat disk in the norm of $C^{1,β},$ $β<1.$

Minimal hypersurfaces scarring along a fixed one in certain manifolds.

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

Paper confirms Yau's conjecture about sphere eigenvalues.