arXiv subjects

Qiang Guang

Publications and source records attributed to Qiang Guang.

10 recordsLinked to original sources

arXiv2022

The $L_p$-Minkowski problem with super-critical exponents

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains widely open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids.

math.AP↗

arXiv2019

Min-max theory for free boundary minimal hypersurfaces II -- General Morse index bounds and applications

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 to prove that for almost every (in the $C^\infty$ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in $M$. If $\partial M$ is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

math.DG↗

arXiv2019

Compactness and generic finiteness for free boundary minimal hypersurfaces (I)

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 of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.

math.DG↗

arXiv2018

Free boundary minimal hypersurfaces with least area

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.

math.DG↗

arXiv2017

Curvature estimates for stable free boundary minimal hypersurfaces

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 a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For the case of $3$-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without any assumption on the area bound. This generalizes Schoen's interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

math.DG↗

arXiv2016

Volume growth, entropy and stability for translating solitons

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give a curvature estimate for translators in $\mathbf{R}^3$ with small entropy. Finally, we estimate the spectrum of the stability operator $L$ for translators and give a rigidity result of $L$-stable translators.

math.DG↗

arXiv2016

On the rigidity of mean convex self-shrinkers

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. Using this result, Colding and Minicozzi were able to establish uniqueness of blowups at cylindrical singularities, and provide a detailed description of the singular set of generic mean curvature flows. In this paper, we show that the bounded curvature assumption is unnecessary for the rigidity of the cylinder if either n is at most 6, or if the mean curvature is bounded below by a positive constant. These results follow from curvature estimates that we prove for strictly mean convex self-shrinkers. We also obtain a rigidity theorem in all dimensions for graphical self-shrinkers, and curvature estimates for translators of the mean curvature flow.

math.DG↗

arXiv2015

Gap and rigidity theorems of $λ$-hypersurfaces

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Second, we show that in one dimension, the only smooth complete and embedded $λ$-hypersurfaces in $\mathbb{R}^2$ with $λ\geq 0$ are lines and round circles. Moreover, we establish a Bernstein type theorem for $λ$-hypersurfaces which states that smooth $λ$-hypersurfaces that are entire graphs with polynomial volume growth are hyperplanes. All the results can be viewed as generalizations of results for self-shrinkers.

math.DG↗

arXiv2015

Rigidity and Curvature Estimates for Graphical Self-shrinkers

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \leq 6$, any smooth, complete self-shrinker $Σ^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on $n$. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow.

math.DG↗

arXiv2014

Self-Shrinkers With Second Fundamental Form of Constant Length

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k \times \mathbb{R}^{2-k}$ for some $k\leq2$. Moreover, we prove a gap theorem for smooth self-shrinkers in all dimensions.

math.DG↗