Stability of Convex Spheres
We prove that strictly convex 2-spheres, all of whose simple closed geodesics are close in length to 2π, are C^0 Cheeger-Gromov close to the round sphere.
math.DG↗
arXiv subjects
Davi Máximo
Publications and source records attributed to Davi Máximo.
Hawking mass and local rigidity of minimal two-spheres in three-manifolds
We study rigidity of minimal two-spheres $Σ$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $Σ$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\times Σ$. We also show that if $Σ$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\mathbb{R}\times\Sph^2$ and can be written as a graph over a slice $Σ_r=\{r\}\times\mathbb{S}^2$, then $Σ$ itself must be a slice, and moreover that slices are indeed local maxima amongst competitors that are graphs with small $C^2$-norm.
math.DG↗
On the blow-up of four dimensional Ricci flow singularities
In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four dimensional manifolds do not necessarily have non-negative Ricci curvature.
math.DG↗
On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian
We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.
math.AP↗