arXiv research

Study systole of large genus minimal surfaces in positive Ricci curvature.

Numerically estimates Colding-Minicozzi entropies of self-shrinkers.

Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.

Study shows rigidity for entropy minimizers in non-monotone cases.

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

Study minimal surfaces in complex hyperbolic space, linking entropy and volume.

Minimal Gaussian surfaces partitioning space with minimum area.

Study self shrinkers with medium entropy in 4D space.

Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

New entropy functionals for curved spaces help predict shape behavior.

New method proves inequalities for self-shrinkers using perturbation.

Following work of Colding-Minicozzi, we define a notion of entropy for connections over $\mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stabilit…

Paper proves entropy conjecture for hypersurfaces and self-shrinkers.

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

No semistability found for Calabi-Yau metrics near cones.

We consider operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $L = Δ+ V +a K.$ Here $Δ$ is the Laplacian of $Σ$, $V$ a non-negative potential on $Σ$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $Σ$ immersed in a Riemannian …

Motivated and inspired by the recent work of Colding [5] and Colding-Minicozzi [6] we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in [5, 6].

We construct Colding-Minicozzi limit minimal laminations in open domains in $\rth$ with the singular set of $C^1$-convergence being any properly embedded $C^{1,1}$-curve. By Meeks' $C^{1,1}$-regularity theorem, the singular set of convergence of a Colding-Minicozzi limit minimal lamination ${\cal L}$ is a locally finit…

In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's res…

We use the Dong-Sogge-Zelditch formula to obtain a lower bound for the volume of the nodal sets of eigenfunctions. Our result improves the recent results of Sogge-Zelditch and in dimensions n \leq 5 gives a new proof for the lower bounds of Colding-Minicozzi.

Minimal Gaussian surface area sets must be round cylinders if they are convex.

The paper studies stability and area growth of $λ$-hypersurfaces.

Entropy for submanifolds in hyperbolic space defined.

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compac…

Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for …

Paper proves compactness and rigidity of $λ$-surfaces in 3D.

Study finds a way to create minimal surfaces with specific properties.

Study ancient caloric functions on graphs, extending a theorem from manifolds.

Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination of U\K. Assuming results of Colding-Minicozzi and an extension …

In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary …

Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable $\mathcal F$-stability. Then, focusing on t…

Let $(M,\bar{g}, e^{-f}dμ)$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compa…

Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.

Proves uniqueness of blowups for forced mean curvature flow.

Study on the formation of singularities in mean curvature flow.

Study proves uniqueness and rigidity of cylindrical self-shrinkers using Łojasiewicz inequalities.

In this paper we prove several results on the geometry of surfaces immersed in $\mathbf R^3$ with small or bounded $L^2$ norm of $|A|$. For instance, we prove that if the $L^2$ norm of $|A|$ and the $L^p$ norm of $H$, $p>2$, are sufficiently small, then such a surface is graphical away from its boundary. We also prove …

New examples of non-bumpy metrics on spheres and projective spaces with multiplicity.

Proves multiplicity one for mean curvature flow singularities.

Study growth rates of harmonic functions on curved surfaces.

The study bounds entropy of plane curves and applies to curve shortening flow.

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

The study disproves rotating ancient flows in 4D space.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

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

The paper shows mean curvature flow keeps diameter bounded under certain conditions.

Uniqueness of conical flows helps understand singularities in surface flows.