Related papers: The Second Variational Formula For the Functional …

For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $\lambda_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $\Delta_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant…

Consider the first nontrivial eigenvalue of the Laplacian on a closed surface as a functional on the space of Riemannian metrics of unit area. N. Nadirashvili has discovered a remarkable connection between critical points of this functional…

For a compact Riemannian $n$-manifold $(M,g)$ of positive scalar curvature, the capital $\Ein$ invariant of $g$ is defined to be the infinimum over $M$ of the quotient of the scalar curvature by the maximal eigenvalue of the Ricci…

This article describes a formula for second variation of generalized Einstein-Hilbert functional on Riemannian manifolds. This work extends the definition of stable Einstein manifolds, and we present some properties.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded…

Recently, the statistical bi-energy functional and its first variational formula were introduced by the author and H. Furuhata. The Maps satisfying the corresponding Euler-Lagrange equation are called statistical biharmonic maps. We present…

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincar\'e-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is…

We study the smallest area $A(M,g)$ of a 2-dimensional stationary integral varifold in a closed Einstein 4-manifold $(M^4,g)$ with $Ric_g = \lambda g, |\lambda|\leq 3, Vol(M,g)\geq v>0, diam(M,g)\leq D, H_1(M;\mathbb{Z})=0.$ Building on the…

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that…

We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional…

Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed…

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a…

We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are…

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe…

Let $M$ be a 5 dimensional Riemannian manifold with $Sec_M\in[0,1]$, $\Sigma$ be a locally conformally flat hypersphere in $M$ with mean curvature $H$. We prove that, there exists $\varepsilon_0>0$, such that $\int_\Sigma (1+H^2)^2 \ge…

Let $(M,g)$ be a pseudo-Riemannian manifold and $F_\lambda(M)$ the space of densities of degree $\lambda$ on $M$. We study the space $D^2_{\lambda,\mu}(M)$ of second-order differential operators from $F_\lambda(M)$ to $F_\mu(M)$. If $(M,g)$…

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\geq6$) Riemannian manifold $(M,g)$ and prove the following results under the condition $\int_{M} \nabla R\cdot\nabla…

We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the…