arXiv research

The study proves properties of self-shrinkers with bounded curvature.

We first consider immersions on compact manifolds with uniform $L^p$-bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J. Langer. In the second part, this result is used to deduce a localized version, being …

Mean curvature flow with uniform bounds on curvature and its gradient

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq κ\leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction fo…

Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.

In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…

In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax the such bound to any finite constant (see Theorem 4.4 for details).

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.

Let $M$ be a compact Riemannian manifold with boundary $\pp M$ and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. Several equivalent statements, including the gradient and Poincaré/log-Sobolev type inequalities of the Neumann semigroup generated by $L$, are presented for lower bound conditions on the curvature of $L$ …

The paper proves rigidity and vanishing theorems for translating solitons.

Improved eigenvalue bounds for minimal hypersurfaces in spheres.

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose…

The study shows that the second fundamental form is intrinsic under certain conditions in space forms.

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary, similar to Shi's local derivative estimates. As an application, we prove a version of …

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…

Let $(M^n,g_0)$ and $(\bar{M}^{n+1},\bar{g})$ be complete Riemannian manifolds with $|\bar{\nabla}^k\bar{Rm}|\le \bar{C}$ for $k \le 2$, and suppose there is an isometric immersion $F_0: M^n \rightarrow \bar{M}^{n+1}$ with bounded second fundamental form. Let $F_t: M^n \rightarrow \bar{M}^{n+1}$ ($t\in [0,T]$) be a fam…

Sharp estimate for genus of embedded surfaces in 3-sphere.

Study classifies 3D self-shrinkers with constant second form norm.

Study bounds on self-shrinkers with bounded HA for applications.

Proves a weak version of Perdomo Conjecture on minimal hypersurfaces.

In this note, we will show a backwards uniqueness theorem of the mean curvature flow with bounded second fundamental form in arbitrary codimension.

Paper classifies special Euclidean hypersurfaces with specific geometric properties.

We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed …

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.

Study on immersions with flat normal bundle in curved spaces.

Sharp estimates for heat flow on nonconvex domains.

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…

An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some characterisations of extrinsic hyperspheres in terms of this curvature are given…

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

An energy estimate is proved for the Bel--Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an $L^{\infty}$ bound on the second fundamental form, and a bound on a spacetime version of Bel--Robinson energy.

This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundam…

Symplectic solitons rigid if bounded, study shows.

The paper generalizes a rigidity theorem for hypersurfaces with constant weighted mean curvature.

We consider critical points of the functionals $Π$ and $Ψ$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion…

The paper studies scalar curvature of self-shrinkers and proves curvature bounds.

Researchers classify special curved spheres in a complex space.

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

The study provides energy estimates for Willmore surfaces and derives a gap statement.

New energy definition for expanding de Sitter spacetime with umbilic boundaries.

In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.

The paper classifies 3D self-expanders with specific properties.

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.