相关论文: First and second variation formulae for the sub-Ri…

We derive a formula for the first variation of horizontal perimeter measure for $C^2$ hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. For $C^2$ hypersurfaces in vertically…

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of…

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring…

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 give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the…

In this paper we compute the first and second variation of the normalized Einstein-Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres.…

Let L be a Lagrangian submanifold of a pseudo- or para-K\"ahler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect…

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of…

In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed…

In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and…

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we…

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $\kappa$. For any point $p\in M$ and any number $\lambda\in\mathbb{R}$ with $\lambda^2+\kappa>0$, we show existence of a $C^2$ spherical surface…

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces…

We prove a generalization of the second variation formula of the Robin function associated to a smooth variation of domains in C^N to the case of the c-Robin function associated to a smooth variation of domains in a complex manifold M…

The stability and the index of complete one-sided minimal surfaces of certain three-dimensional Riemannian manifolds with positive scalar curvature are studied.

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

This paper examines minimal hypersurfaces in sub-Riemannian Heisenberg groups. We extend the celebrated Simons formula and Kato inequality to the sub-Riemannian setting, and we apply them to obtain integral curvature estimates for stable…

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 prove new variation formulae for the volume of coassociative submanifolds, expressed in terms of $G_2$ data. As a special case, we obtain a second variation formula for variations within the moduli space of coassociative submanifolds;…