arXiv research

Estimates for $p$-capacities on symmetric manifolds.

Paper proves March's criterion for transience on symmetric manifolds.

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.

Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.

3-manifold curvature comparison with rotationally symmetric bodies.

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…

The paper classifies hypersurfaces in Heisenberg groups with rotational symmetry.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…

New findings on magnetic geodesic flows and periodic motions.

New theorems on compactness and finiteness for specific types of self-shrinkers.

New Ricci flow solutions found with rotational symmetry and cone-like singularities.

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones a…

Sharp inequalities found for orbifold metrics.

We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.

We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal h…

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.

In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to $-\frac{1}{r}$, where $r$ stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rota…

New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.

Researchers set entropy limits for specific types of self-shrinkers.

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold $M = (0,\infty) \times Y$ whose rotation radius is constant outside some compact interval. The Laplacian on $M$ is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…

The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give a…

Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.

The paper classifies 3D complete gradient Yamabe solitons.

The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.

In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries a…

Extends Milnor's criterion to biharmonic functions.

The paper proves existence and instability of weak $r$-harmonic maps.

New method to bound Laplacian eigenvalues of geodesic balls.

Notes on Frenet-Serret formulas for curves in flat pseudo-hermitian manifolds.

We consider 3+1 rotationally symmetric Lorentzian Einstein spacetime manifolds with $Λ>0$ and reduce the equations to 2+1 Einstein equations coupled to `shifted' wave maps. Subsequently, we prove various (explicit) positive mass-energy theorems. No smallness is assumed.

Proves conjecture about sphere widths under rotational symmetry.

The flow of a torus by inverse mean curvature keeps total curvature bounded until singularity.

Variational approximations for curve flows on Riemannian manifolds.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

Study shows flows from double cones remain symmetric, finds non-symmetric example.

Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

In this paper we study the classification of ancient convex solutions to the mean curvature flow in $\R^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a …

New minimal tori found in curved spaces.

In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We als…

We obtain supremum of the k-th normalized Steklov eigenvalues of all rotational symmetric conformal metrics on the cylinder with k>1. The case k=1 for all conformal metrics has been completely solved by Fraser and Schoen. We give geometric description in terms of minimal surfaces for metrics attaining the supremum. We …

Upper bound on index of rotationally symmetric self-shrinking tori.