微分几何

We extend the Otal-Rosas bound on the number of small eigenvalues of the Laplacian on a hyperbolic surface to the small eigenvalues of pseudo-Laplacians. In the process, we extend the work of Colin de Verdi\`ere on the spectral theory of…

For a $k$-dimensional Brakke flow on an open subset $U \subset \mathbf{R}^{n}$, over an open time interval $J$, we prove the existence of a canonical space-time-Grassmann measure $\lambda$, over $J \times \mathbf{G}_{k} (U)$, and give a…

This is a monograph devoted to the study of Kirwan's real polytopes, i.e., in the context where there are involutions on the group and on the symplectic manifold. This work brings together and completes the two preprints I had already…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

We study oriented Riemannian $4n$-manifolds whose Thorpe $2n^{\text{th}}$ curvature operator $\hat{R}_{2n}\colon\Lambda^{2n} \longrightarrow \Lambda^{2n}$, or its Weyl analogue $\hat{W}_{2n}$, commutes with the Hodge star. For pure…

We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. Under the natural inequality condition on the…

For a vector bundle $E^{n+k}$ over a closed manifold $M^n$ with $k$ even and $n$ odd, we equip the metric with an adiabatic parameter, and prove that the index of $E$ is the same as the index of $M$. We also introduce an analog of analytic…

We define a mass function on asymptotically hyperbolic manifolds with continuous metrics via the normalized Ricci DeTurck flow. This definition coincides with the classical mass for smooth metrics. We also introduce the scalar curvature…

This paper provides a construction and existence proof for a 1-parameter family of chiral unbalanced triply-periodic minimal surfaces of genus 4. We name these {\textit{gyrating H'-T} surfaces, because they are related to Schoen's H'-T…

We show that any toric asymptotically conical shrinking gradient K\"ahler-Ricci soliton on an anti-canonically polarised resolution of a K\"ahler cone satisfies a complex Monge-Amp\`ere equation. We then set up an Aubin continuity path to…

A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form. We address the same…

In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator…

For a one-parameter degeneration of compact Riemann surfaces endowed with the K\"ahler metric induced from the K\"ahler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a…

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…

In this paper, we provide necessary and sufficient conditions for the existence of para-Kaehler immersions in para-Kaehler space forms. As a consequence, we prove that, in general, a local para-Kaehler immersion cannot be globally extended,…

We provide a generalization of global hyperbolicity in pseudo-Riemannian spaces of signature (p, q) for p ___ q ___ 2. We then prove the compactness of causal diamonds in globally hyperbolic spaces and deduce the existence of solutions to a…

We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded…

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g…

For covering spaces and properly discontinuous actions with compatible diffusion processes, we discuss Lyons-Sullivan discretizations of the processes and the associated function theory.

We prove some rigidity and classification results for graphs with prescribed mean curvature and locally constant Dirichlet and Neumann data, for instance as they appear in capillarity problems. We consider domains in Riemannian manifolds,…