Prescribing Ricci curvature on homogeneous spaces

The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M = G / K {M=G/K} is studied. We focus on the metrics at which the map g ↦ Rc ⁡ ( g ) {g\mapsto\operatorname{Rc}(g)} is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.

A conformal Yamabe problem with potential on the Euclidean space

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

Continuum of one-sign solutions of one-dimensional Minkowski-curvature problem with nonlinear boundary conditions

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.