Programação:
14:00 - 14:50: Matheus Hudson (UFSCAR) - Mean Curvature Flow in an Extended Ricci Flow Background.
15:00 - 15:50: André Gomes (IMECC) - On Rigidity of Horizontal Curvature Bounds of Isometric Actions.
16:00 - 16:20: Café
16:20 - 17:10: João Henrique Andrade (IME-USP) - Complete metrics with constant fractional higher order Q-curvature on the punctured sphere.
Mean Curvature Flow in an Extended Ricci Flow Background
Matheus Hudson- (UFSCAR)
In this talk, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying mean curvature flow in a Ricci flow background. Mainly, the functional we focus on the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties, from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman's modified extended Ricci flow. In this time-derivative formula an extension of Hamilton's differential Harnack expression on the boundary integrand appears. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Finally, we present examples of mean curvature solitons in an extended Ricci flow background.
On Rigidity of Horizontal Curvature Bounds of Isometric Actions singularities
André Gomes (IMECC)
Let $G\curvearrowright M$ be an isometric action of a Lie Group on a complete orientable Riemannian manifold. We disintegrate absolutely continuous measures with respect to the volume measure of $M$ along the principal orbits of $G\curvearrowright M$ and define a functional on the probability measures with support on the principal orbits of the action to further prove that the convexity properties of this functional guarantees necessary and sufficient conditions to the Ricci curvature of $M$ on horizontal directions to be bound below by a given real number $K$.
Complete metrics with constant fractional higher order Q-curvature on the punctured sphere.
João Henrique Andrade (IME-USP) -
This talk is devoted to constructing complete metrics with constant higher fractional curvature on punctured spheres with finitely many isolated singularities. Analytically, this problem is reduced to constructing singular solutions for a conformally invariant integro-differential equation that generalizes the critical GJMS problem. Our proof is based on a gluing method, which we briefly describe. Our main contribution is to provide a unified approach for fractional and higher order cases. This method relies on proving Fredholm properties for the linearized operator around a suitably chosen approximate solution. The main challenge in our approach is that the solutions to the related blow-up limit problem near isolated singularities need to be fully classified; hence we are not allowed to use a simplified ODE method. To overcome this issue, we approximate solutions near each isolated singularity by a family of half-bubble tower solutions. Then, we reduce our problem to solving an (infinite-dimensional) Toda-type system arising from the interaction between the bubble towers at each isolated singularity. Finally, we prove that this system's solvability is equivalent to the existence of a balanced configuration. This is a joint work with J. Wei and Z. Ye.