Related papers: Riemannian $\lambda_1$-extremal Metrics on General…

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor on $X$ such that $K_X+D$ is big and nef. We first prove that the singular K\"ahler--Einstein metric constructed by Berman--Guenancia is almost-complete on $X…

Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the…

Let (X,[\omega]) be a compact Kaehler manifold with a fixed Kaehler class [\omega]. Let K_\omega be the set of all Kaehler metrics on X whose Kaehler class equals [\omega]. In this paper we investigate the critical points of the functional…

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For…

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface \Sigma in a compact n-manifold M, there is a Riemannian…

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and…

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we…

In this paper, I describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. I describe also all the limits with…

Let $(M^4,g)$ be a closed Riemannian manifold of dimension four. We investigate the properties of metrics which are critical points of the eigenvalues of the Paneitz operator when considered as functionals on the space of Riemannian metrics…

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq…

We give a set of sufficient and necessary conditions for parabolicity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.

We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and…

On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kaehler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for…

We glue two manifolds which have curvature operators at least k (in the sense of eigenvalues) along their common boundary. We show that if the sum of the second fundamental forms of the boundary is positive semidefinite, then the curvature…

Let G be a complex semisimple Lie group, K a maximal compact subgroup and V an irreducible representation of K. Denote by M the unique closed orbit of G in P(V) and by O its image via the moment map. For any measure on M we construct a map…

We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal K\"ahler metrics on a compact K\"ahler manifold introduced in our previous work. This extends a result by…

We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^\infty (M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric…

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature K\"ahler metrics to the case of extremal metrics. Namely, we show that a given…

The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite…

In this paper we are dealing with quantitative Rellich inequalities on Finsler-Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on…