相关论文: Conformal change of Riemannian metrics and biharmo…

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a…

We prove that certain subcritical Paneitz-Branson type equations on a closed Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positIve solutions, where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This implies…

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian…

A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and…

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main…

Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures…

We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric…

We study the Lorentzian Calder\'on problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This…

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is…

Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution $X$ admits a large set of ``adapted" metrics in $M$ for which $X$ solves the corresponding…

We consider a conformally invariant version of the Calder\'on problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main…

We introduce Condition W $\,$(1.2) for a smooth differential form $\omega$ on a complete noncompact Riemannian manifold $M$. We prove that $\omega$ is a harmonic form on $M$ if and only if $\omega$ is both closed and co-closed on $M\, ,$…

In this paper, we study the subcritical biharmonic equation \[\Delta ^2 u=u^\alpha\] on a complete, connected, and non-compact Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature. Using the method of invariant tensors, we derive…

We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature…

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent…

For a compact, connected, oriented Riemannian $3$-manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give a local representation and a full symbol expression for the electromagnetic Dirichlet-to-Neumann map by factorizing…

Given a symmetric Riemannian manifold (M, g), we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number {\epsilon}…

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq…

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a…

The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin,…