arXiv research

Minimal surfaces in 8D smooth and nondegenerate.

Smooth minimizing hypersurfaces in 11D are generic, with singularities in higher dimensions.

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

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

Generic smooth minimal hypersurfaces exist in 8D manifolds.

Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.

In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in sca…

The paper constructs metrics on spheres with families of minimal hypersurfaces.

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…

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted vo…

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 …

Paper proves properties of minimal hypersurfaces in specific solitons.

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

Paper estimates curvature of minimal surfaces in a specific geometric space.

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…

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 …

Theorem proves minimal hypersurfaces in nonnegative scalar curvature manifolds are smooth.

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

Study on smoothness of 4D Willmore-type hypersurfaces.

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

The study finds generic regularity of minimal hypersurfaces in Riemannian manifolds.

Minimal surfaces' boundary points are always smooth.

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

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

The paper proves the existence of $G$-invariant minimal hypersurfaces on certain Riemannian manifolds.

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to any $n$.

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.

Proves existence of a single-valued minimal hypersurface in compact manifolds.

The study proves that certain minimal surfaces are flat under specific conditions.

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

Study on stable minimal hypersurfaces under Ricci curvature constraints.

Survey solves curvature problems with hyperbolic spaces.

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifo…

New stable minimal hypersurfaces found in 4-manifolds, proving topology results.

Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then $M$ must be an affine hyperplane.

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.

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…

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

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …

Study min-max theory for hypersurfaces with boundary constraints.

Smooth approximations near singularities of constant mean curvature surfaces are found.

Optimal regularity theory for stable minimal hypersurfaces with small singular set.

Compact hypersurfaces minimize area in convex cones with free boundary.

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

Smooths out complex shapes into simpler forms.

We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space with an anisotropic energy. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface. The minimizer of such an energy among all closed hy…

In [5], Colding-Ilmanen-Minicozzi-White showed that within the class of closed smooth self-shrinkers in $\mathbb{R}^{n+1}$, the entropy is uniquely minimized at the round sphere. They conjectured that, for $2\leq n\leq 6$, the round sphere minimizes the entropy among all closed smooth hypersurfaces. Using an appropriat…