Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

We present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem

We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space $$(X,d_X)$$ ( X , d X ) . This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold $$S\subset {{\mathbb {R}}}^m$$ S ⊂ R m , $$m>n$$ m > n needs to be constructed to approximate a point cloud in $${{\mathbb {R}}}^m$$ R m . These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in $${{\mathbb {R}}}^m$$ R m and interpolated to a smooth submanifold.

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

Embolic aspects of black hole entropy

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in [Formula: see text]-dimensional spacetimes. We treat the case of horizons having space-like sections [Formula: see text] which are topological spheres, following Hawking’s and the Topological Censorship theorems. We use the injectivity radius of the induced metric on [Formula: see text] to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of [Formula: see text] as the fundamental quantity expressing the complexity of [Formula: see text] on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.