Numerical computation of the cut locus via a variational approximation of the distance function

We propose a new method for the numerical computation of the cut locus of a compact submanifold of R3 without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of R3.

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃 m

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f.

HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

SOME SPHERE THEOREMS FOR SUBMANIFOLDS WITH POSITIVE BIORTHOGONAL CURVATURE

The purpose of this paper is to investigate sphere theorems for submanifolds with positive biorthogonal (sectional) curvature. We provide some upper bounds for the full norm of the second fundamental form under which a compact submanifold must be diffeomorphic to a sphere.

Some results on Finsler submanifolds

In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

An inclination lemma for normally hyperbolic manifolds with an application to diffusion

Let ($M$, ${\rm\Omega}$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^{l}$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^{1}$-submanifold ${\rm\Delta}$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_{N}M$. We suppose that ${\rm\Delta}$ transversely intersects the stable manifold of $N$. Then, we prove that for all ${\it\varepsilon}>0$, and for $n\in \mathbb{N}$ large enough, there exists $x_{n}\in N$ such that $f^{n}({\rm\Delta})$ is ${\it\varepsilon}$-close, in the $C^{1}$ topology, to the strongly unstable manifold of $x_{n}$. As an application of this ${\it\lambda}$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold’s example).

Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K ⊂ M so that the outward normal exponential map off the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well-defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

First eigenvalue of submanifolds in Euclidean space

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.