Giona Veronelli

Topology of steady and expanding gradient Ricci solitons via f-harmonic maps

Abstract In this paper we give some results on the topology of manifolds with ∞-Bakry–Emery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f -harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yauʼs theory of harmonic maps.

REMARKS ON Lp-VANISHING RESULTS IN GEOMETRIC ANALYSIS

We survey some Lp-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrodinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed.

Sobolev spaces of maps and the Dirichlet problem for harmonic maps

In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S. Hildebrandt, H. Kaul and K. Widman in 1977. In particular no curvature assumptions on the target are required. Our proof relies on a careful analysis of the Sobolev spaces of maps involved in the variational process, and on a deformation result which permits to glue a suitable Euclidean end to the geodesic ball.

Hyperbolization of cusps with convex boundary

We prove that for every metric on the torus with curvature bounded from below by −1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem: every metric with curvature bounded from below on a compact surface is isometric to a convex surface in a 3-dimensional space form.

Lower volume estimates and Sobolev inequalities

We consider complete manifolds with asymptotically non-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an explicit lower control on the volume of geodesic balls. In case the amount of negative curvature is small and the Sobolev constant is almost optimal, we deduce that the manifold is diffeomorphic to Euclidean space. This extends previous results by M. Ledoux and C. Xia.

Boundary structure of convex sets in the hyperbolic space

We prove some results concerning the boundary of a convex set in

Quantitative regularity for p-harmonic maps

\mathbb {H}^n

Uniform decay estimates for finite-energy solutions of semi-linear elliptic inequalities and geometric applications

Hn. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian curvature measure.