Related papers: Riemannian manifolds with maximal eigenfunction gr…

Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all…

In this article we study any 4-dimensional Riemannian manifold $(M,g)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the following equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+…

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly…

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive…

Let M be a compact, connected, m-dimensional manifold without boundary and p>1. For 1<p\leq m, we prove that the first eigenvalue \lambda_{1,p} of the p-Laplacian is bounded on each conformal class of Riemannian metrics of volume one on M.…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. For a metric $g$ on $M$, we let $\la_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:= \frac{4(n-1)}{n-2} \Delta_g + \scal_g$. Then, the second Yamabe invariant…

In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such…

Let $\mathcal{C}(\mathcal{R},n,p,\Lambda,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\leq D$, non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvature condition…

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem…

Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $\varphi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h…

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where $\Delta_p$ is the $p-$laplacian, with $1<p<n$. The…

We study complete, connected and simply connected $n$-dim Riemannian manifold $M$ satisfying Ricci curvature lower bound. Further more, suppose that $M$ admits discrete isometric group actions $G$ so that the diameter of the quotient space…

We study the approximation of functions that map a Euclidean domain $\Omega\subset \mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\subset W^{1,2}(\Omega,M)$.…

For a closed, connected direct product Riemannian manifold $(M, g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus \cdots\oplus f_l^2g_l\}$ of all…

Given a closed Riemannian manifold (M, g), there is a partition \Sigma_i of its tangent bundle TM called the focal decomposition. The sets \Sigma_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there…

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M, g)$ from a restriction $\Lambda_{\Src, \Rec}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\Src$ and…