arXiv subjects

Isaac M. Lopez

Publications and source records attributed to Isaac M. Lopez.

4 recordsLinked to original sources

arXiv2025

Dynamical functionals on ancient ARF Ricci flows

We introduce a dynamical energy functional on compact ancient asymptotically Ricci-flat Ricci flows with modest decay using limits of conjugate heat flows. This functional satisfies a steady Ricci breather-type rigidity and provides an upper bound for the ordinary $λ$-functional while retaining many of its properties. In addition, motivated by work of Colding and Minicozzi, we derive local eigenvalue estimates for normalized Ricci flows coupled with conjugate heat flows.

math.DG↗

arXiv2024

A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier-Ricci curvature

We prove an inequality relating the isoperimetric profile of a graph to the decay of the random walk total variation distance $\sup_{x\sim y} ||P^n(x,\cdot)-P^n(y,\cdot)||_{\mathrm{TV}}$. This inequality implies a quantitative version of a theorem of Salez (GAFA 2022) stating that bounded-degree graphs of non-negative Ollivier-Ricci curvature cannot be expanders. Along the way, we prove universal upper-tail estimates for the random walk displacement $d(X_0,X_n)$ and information $-\log P^n(X_0,X_n)$, which may be of independent interest.

math.PR↗

arXiv2024

Ancient and expanding spin ALE Ricci flows

We classify spin ALE ancient Ricci flows and spin ALE expanding solitons with suitable groups at infinity. In particular, the only spin ancient Ricci flows with groups at infinity in $SU(2)$ and mild decay at infinity are hyperkähler ALE metrics. The main idea of the proof, of independent interest, consists in showing that the large-scale behavior of Perelman's $μ$-functional on any ALE orbifold with non-negative scalar curvature is controlled by a renormalized $λ_{\mathrm{ALE}}$-functional related to a notion of weighted mass.

math.DG↗

arXiv2024

Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces

We establish a weighted positive mass theorem which unifies and generalizes results of Baldauf--Ozuch and Chu--Zhu. Our result is in fact equivalent to the usual positive mass theorem, and can be regarded as a positive mass theorem for smooth metric measure spaces. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.

math.DG↗