We prove $L^p$-bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.

We give an introduction into and exposition of Seiberg-Witten theory.

Following joint work with Dyatlov [DyGu], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of ${ℝ}^d$, under the condition that the geodesic flow has trapped set $K$ of Liouville measure $0$.

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.

We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.

We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.

Let (M,g) be a compact Riemannian manifold without boundary, with dim M ≥ 3, and f: ℝ → ℝ a continuous function which is sublinear at infinity. By various variational approaches, existence of multiple solutions of the eigenvalue problem $-{\Delta }_g\omega +\alpha \left(\sigma \right)\omega =K̃(\lambda ,\sigma )f\left(\omega \right)$, σ ∈ M, ω ∈ H₁²(M), is established for certain eigenvalues λ > 0, depending on further properties of f and on explicit forms of the function K̃. Here, ${\Delta }_g$ stands for the Laplace-Beltrami operator on (M,g), and α, K̃ are smooth positive functions. These multiplicity...

Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L={\int }_{\Sigma }\frac{1}{cos\alpha }d\mu$ in the class of symplectic surfaces. It is ${cos}^3\alpha H={\left(J{(J\nabla cos\alpha )}^{\top }\right)}^{\perp }$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative...

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space $H_2^p(M,T^{*}M\otimes T^{*}M)$ with $p\>\frac{n}{2}$ and there exist a point $P\in M$ and $\delta \>0$ such that $g$ is smooth in the ball $B_p\left(\delta \right)$. We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to $g$ and of volume $1$. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with...

We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide,...