arXiv subjects
Publications and source records attributed to Antonio G. Ache.
Consider a sample of $n$ points taken i.i.d from a submanifold $Σ$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $Σ$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carré du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis.
We consider a normalization of the Ricci flow on a closed Riemannian manifold given by the evolution equation $\partial_{t}g(t)=-2(Ric(g(t))-\frac{1}{2τ}g(t))$ where $τ$ is a fixed positive number. Assuming that a solution for this equation exists for all time, and that the full curvature tensor and the diameter of the manifold are both uniformly bounded along the flow, we prove that up to the action of a 1-parameter family of diffeomorphisms, the flow converges to a unique shrinking gradient Ricci soliton using an idea of Sun and Wang to study the stability of the Käler Ricci flow near a Kähler-Einstein metric. This idea relies on the monotonicity of the Ricci flow along Perelman's $\W$-functional and a Łojasiewicz-Simon inequality for the $μ$-functional. Our result is an extension of the main theorem in Sesum (2006), where convergence to a unique soliton is proved assuming that the solution of the normalized Ricci flow in consideration is sequentially convergent to a soliton which satisfies a certain integrability condition.
We analyze the indicial roots of the self-dual deformation complex on a cylinder $(\mathbb{R} \times Y^3, dt^2 + g_Y)$, where $Y^3$ is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section $Y^3$. We also resolve a conjecture of Kovalev-Singer in the case where $Y^3$ is a hyperbolic rational homology 3-sphere, and show that there are infinitely many examples for which the conjecture is true, and infinitely many examples for which the conjecture is false. Applications to gluing theorems are also discussed.