The Hopf Boundary Point Lemma for Vector Bundle Sections

In real analysis, the Lojasiewicz inequalities, revitalized by Leon Simon in his pioneering work on singularities of energy minimizing maps, have proven to be monumental in differential geometry, geometric measure theory, and variational problems. These inequalities provide specific growth and stability conditions for prescribed real-analytic functions, and have found applications to gradient flows, gradient systems, and as explicated in this paper, vector bundles over compac...

In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize the result of Hopf as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reis...

This is the first part of the lecture notes that grew out of the special course given during the 2021-2022 academic year. In these lecture notes we present an approach to the fundamental structures of differential geometry that uses the vernacular of sheaves, differential operators and horizontal subbundles.

We generalize a theorem by J. Choe on capillary surfaces for arbitrary 3-dimensional spaces of constant curvature. The main tools in this paper are an extension of a theorem of H. Hopf due to S.-S. Chern and two index lemmas by J. Choe.

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.

The algorithm for inverting covariant exterior derivative is provided. It works for a sufficiently small star-shaped region of a fibered set - a local subset of a vector bundle and associated vector bundle. The algorithm contains some constraints that can fail, giving no solution, which is the expected case for parallel transport equations. These constraints are straightforward to obtain in the proposed approach. The relation to operational calculus and operator theory is out...

In this paper, we investigate the mean curvature flow having equifocal submanifolds as initial data. The investigation are performed by investigating the mean curvature flow having the lifted submanifolds to a Hilbert space through a Riemannian submersion as initial data.

The sub-Finslerian geometry means that the metric $F$ is defined only on a given subbundle of the tangent bundle, called a horizontal bundle. In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-Finslerian manifolds, which relates the properties of completeness, geodesically completeness, and compactness. The sub-Finsler bundle, the exponential map and the Legendre transformation are deeply involved in this investigation.

We study the existence and regularity of invariant graphs for bundle maps (or bundle correspondences with generating bundle maps motivated by ill-posed differential equations) having some relative partial hyperbolicity on non-trivial and non-locally-compact bundles. The regularity includes (uniform) $ C^0 $ continuity, H\"older continuity and smoothness. To illustrate the power of our results and methods, a number of applications to both well-posed and ill-posed semilinear di...

In this paper, we consider different versions of the classical Hopf's boundary lemma in the setting of the fractional $p-$Laplacian for $p \geq 2$. We start by providing for a new proof to a Hopf's lemma based on comparison principles. Afterwards, we give a Hopf's result for sign-changing potential describing the behavior of the fractional normal derivative of solutions around boundary points. The main contribution here is that we do not need to impose a global condition on t...