arXiv research

Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature esti…

Sharp area estimates for minimal submanifolds in curved spaces.

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

Estimates submanifold diameters in curved spaces.

We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.

Sampling random points can reveal submanifold topology.

Estimates spectral projections restricted to uniformly embedded submanifolds.

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into…

Upper bounds for essential spectrum of minimal submanifolds linked to volume growth.

The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lowe…

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $\mathcal{A}$-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that …

Estimates the first eigenvalue of a Schrödinger operator on minimal submanifolds.

We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yie…

In this paper, we analyze the geometric structure of an Euclidean submanifold whose osculating spaces form a nonconstant family of proper subspaces of the same dimension. We prove that if the rate of change of the osculating spaces is small, then the submanifold must be a (submanifold of a) ruled submanifold of a very …

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

Study mean curvature flow of high codimension submanifolds in complex projective space.

Derives an inequality for submanifolds in spheres.

In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for $4$-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space, the corresponding estimates for the first eigenvalue are also obtained.

We extend the estimate obtained in [1] for the mean curvature of a cylindrically bounded proper submanifold in a product manifold with an Euclidean space as one factor to a general product ambient space endowed with a warped product structure.

Optimizes eigenvalue bounds for submanifold Dirac operators.

Rigidity for 4D Willmore submanifolds with boundary.

We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n \times_f Q^q$, where $f \in C^\infty(N)$. We also study the essential spectrum of these minimal submanifolds.

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …

We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the…

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifol…

Study shows submanifolds can't be immersed in certain spaces.

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same $\mathbb{Z}/2$ homology class in $\mathbb{C}P^n$ (but not among all submanifolds in this…

In this paper, we estimate the eigenvalues of the twisted Dirac operator on Kähler submanifolds of the complex projective space $CP^m$ and we discuss the sharpness of this estimate for the embedding $CP^d \hookrightarrow CP^m$.

Paper proves new isoperimetric inequality for minimal submanifolds with free boundary.

Sharp $L^p$-logarithmic-Sobolev inequalities on submanifolds with applications to hypercontractivity.

New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.

Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous result…

We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for the Bernstein type theorem.

Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates fo…

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional cur…

We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\ir{n+m},\, m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean spaces of arbitrary codimension.

Burq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be remov…

In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $\varphi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…

Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $N\times\mathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fund…

Study optimizes decay estimates for minimizing currents in submanifolds.

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as cons…

Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Lapl…

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge-de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenv…

New method uses manifold learning to infer latent positions of 1D submanifolds in random dot product graphs.

High codimension submanifolds evolve to convex shapes, leading to smooth limiting flows.