arXiv research

The paper classifies and studies conformal variations of submanifolds.

Study variational submanifolds in Euclidean spaces from dynamical systems.

New formulas for coassociative submanifolds' volume variation.

The paper studies variational functionals for submanifolds using the Lepage form.

The paper studies variations of $σ_u$-curvature for submanifolds in Riemannian manifolds.

The paper studies infinitesimal variations of submanifolds in Euclidean space.

We study the geometry of jets of submanifolds with special interest in the relationship with the calculus of variations. We give a new proof of the fact that higher order jets of submanifolds are affine bundles; as a by-product we obtain a new expression for the associated vector bundles. We use Green-Vinogradov formul…

The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.

The paper studies submanifolds of fixed degree with constraints on variations.

Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.

We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.

Minimal submanifolds are found as energy concentration sets in variational problems.

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 $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in compl…

We give asymptotically tight estimates of tangent space variation on Riemannian submanifolds of Euclidean space with respect to the local feature size of the submanifolds. We show that the result follows directly from structural properties of local feature size of the Riemannian submanifold and some elementary Euclidea…

First variation of fractional $k$-dimensional measure for submanifolds

The paper studies variations of weighted curvature on submanifolds.

A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincare-Einstein space with prescribed…

The paper defines and studies new types of submanifolds in a unit sphere.

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation $H+σ=0$, where $H$ is analogous to the…

Variational characterization of calibrated submanifolds in different contexts.

Study improves understanding of submanifold reach in Riemannian geometry.

Study $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.

The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian. This problem in the general version remains unsolved. Here, we contribute to it w…

We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lacking regularity, several non-smooth extensions of the result are discussed. A…

Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.

In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.

Paper calculates second variation of Graham-Witten energy for spheres.

We extend the notion of $r$-minimality of a submanifold in arbitrary codimension to $u$-minimality for a multi-index $u\in\mathbb{N}^q$, where $q$ is the codimension. This approach is based on the analysis on the frame bundle of orthonormal frames of the normal bundle to a submanifold and vector bundles associated with…

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler 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 to Hamiltonian variations. We apply this formula to several cases. In particular …

$ξ$-submanifold in the Euclidean space $\bbr^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $\bbr^{m+p}$. It is also a generalization of the $λ$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $ξ$-submanifolds ar…

New method calculates cut locus on surfaces without boundary.

Develops geometric inverse problem for discrete mechanics.

The paper proves inequalities for submanifolds in Riemannian manifolds.

There is a Lorenzian group acting on the conformal space ${\mathbb Q}^n_p$. We study the regular submanifolds in the conformal space ${\mathbb Q}^n_p$ and construct general submanifold theory in the conformal space ${\mathbb Q}^n_p$. Finally we give the first variation formula of the Willmore volume functional of subma…

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of th…

This paper surveys some of the known results on $δ$-ideal CR submanifolds in complex space forms, the nearly Kähler $6$-sphere and odd dimensional unit spheres. In addition, the relationship between $δ$-ideal CR submanifolds and critical points of the $λ$-bienergy is mentioned. Some topics on variational problem for th…

Study adiabatic limits of calibrated submanifolds in Riemannian geometry.

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebr…

This paper develops a method to learn lower-dimensional submanifolds of brain connectomes.

We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $φ$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $φ$-volume to obtain the stability result in so…

Hasse principle applied to area-minimizing submanifolds across different homology types.

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a ge…

A Hamiltonian stationary Lagrangian submanifold of a Kaehler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kaehler manifold of real dimension four that guarantees the existence of a family of small Hamiltonian stationar…

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of the spacetime, the completeness of suitable Finsler metrics associated to it an…

On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the …

We study underlying geometric structures for integral variational functionals, depending on submanifolds of a given manifold. Applications include (first order) variational functionals of Finsler and areal geometries with integrand the Hilbert 1-form, and admit immediate extensions to higher-order functionals.

Smoothness of Hamiltonian stationary submanifolds in symplectic manifolds proven.