Multiplicity one for min-max theory in compact manifolds with boundary and its applications

Morse index of multiplicity one min-max minimal hypersurfaces

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-index}, \cite{neves-icm}) proposed by the authors with the goal of developing a Morse theory for the area functional.

Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature

We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis).…

边界处任意重数面积最小化流的Allard型正则性理论

Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicity

本文推广了Allard边界正则性定理，研究具有光滑边界且边界取任意重数的面积最小化流，并由此得出重要结构性结论。

关于双曲曲面的第一宽度：重数与下界

On the first width of hyperbolic surfaces: multiplicity and lower bounds

该研究刻画了负曲率闭黎曼曲面上单参数Allen-Cahn极小极大构造的闭测地线构型，证明测地线重数为1，得到闭双曲曲面第一极小极大宽度的一致锐下界，还计算了Bolza曲面等的第一宽度。

在给定同调类中等价min-max理论的单重性

Multiplicity one for equivariant min-max theory in prescribed homology classes

本文研究了在给定同调类中等价min-max理论的单重性，并证明了存在无限多个$G$-不变极小超曲面，同时建立了具有$G$-索引上界的等价min-max理论。

Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen--Cahn

In Guaraco's 2018 work a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold $N^{n+1}$ with $n\geq 2$. This was achieved by employing an Allen--Cahn approximation scheme and a one-parameter minmax for the Allen--Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to $0$). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we consider a minmax construction that is a modification of the one in the aforementioned work, by allow…

Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of th…

A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal hypersurfaces and a pointwise curvature estimate for the hypersurfaces in this class in low dimensions are also discussed.

极小极大测地线的指标、交点与重数

Index, Intersections, and Multiplicity of Min-Max Geodesics

本文证明了闭曲面上实现 $p$-宽度的极小极大测地线的 Morse 指标与交点个数上界，并构造 $S^2$ 上度量开集说明重数一不通用成立。

闭流形中具有重数$2$收敛性的平均曲率流

Mean curvature flow with multiplicity $2$ convergence in closed manifolds

本文在闭流形中构造了光滑嵌入连通超曲面的不朽平均曲率流新例子，证明其随时间趋于无穷以重数2收敛到极小超曲面。

任意重数下面积最小化流在边界处切锥的唯一性

On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity

本文证明任意整数边界重数的面积最小化流在密度 $Q/2$ 边界点处切锥唯一且收敛具有幂次速率，并由此导出支撑高重数边界正则性理论的分解定理。

Gross-Pitaevskii方程在黎曼流形上的解的多重性

Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds

本文研究了在黎曼流形上Gross-Pitaevskii方程解的多重性问题，通过临界点理论和Γ收敛方法，结合新的等周型问题结果，得出了解的多重性下界。

Multiplicity Theorems for Frechet Manifolds

We prove multiplicity theorems for Keller $ C_c^1 $-functionals on Frechet spaces and Finsler manifolds which are invariant under the action of a discrete subgroup. For such functionals, we evaluate the minimal number of critical points by applying the Lusternik-Schnirelmann category.

Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in $\mathbb R^3$ modeled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding-Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in $\mathbb R^3$ modeled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbations of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-kno…

Existence of hypersurfaces with prescribed mean curvature I - Generic min-max

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 hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including smooth Mor…

Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann theory

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + δh)$ has at least $Cat(M) +1 $ solutions for $δ$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-category of $M$. Cat(M) of the solutions obtained have energy arbitrarily close to the minimum.