Global minimizers exist for Tonelli Lagrangians on half-Lie groups.
problem Existence and properties of minimizers for Lagrangians on infinite-dimensional spaces.
method Introduced Tonelli Lagrangians on half-Lie groups, proved existence of minimizers and flow lines.
result Global minimizers exist above certain energy thresholds.
New curvature measure for optimal transport with specific cost function.
problem Optimal transport with specific cost function.
method Proposed generalized curvature measure.
result Non-negativity of the generalized curvature implies displacement convexity.
We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the n-dimensional torus that have no conjugate points are C0 integrable, i.e. $T^*\T^n$ is C0 foliated by a family $\Fc$ of invariant C0 Lagrangian graphs. Assuming that the Hamiltonian is C∞, we prove that there …
New orbits found in Lagrangian systems on surfaces.
problem Finding action minimizing periodic orbits in Tonelli Lagrangian systems.
method Analyzing minimal boundaries and using graph theorems.
result Existence of action minimizing simple periodic orbits.
An example from Almgren and Federer shows geodesics that are not always the shortest.
problem Illustrating the subtleties of geodesic minimization in complex metrics.
method Exposition of a specific example in S1imesS2 to clarify definitions. result Found geodesics that are not minimizers in their homotopy classes.
The study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr,r>1, where d is the Riemannian distance of a complete …
Study on singularities of solutions to Hamilton-Jacobi equations on manifolds.
problem Characterizing singularities of solutions to time-dependent Hamilton-Jacobi equations.
method Uniformly continuous viscosity solutions, Tonelli Hamiltonians, homotopy theory.
result The set of points where solutions are not differentiable is locally contractible.
Smooth maps bound Betti numbers of zero sets.
problem Bounding Betti numbers of zero sets of smooth maps.
method Generalized Thom-Milnor bound to polynomial maps on nonsingular real algebraic varieties; introduced condition number for families of functions.
result Extended Thom-Milnor bounds to families of functions and semialgebraic sets.