arXiv research

Given a parallel calibration $φ\in Ω^p(M)$ on a Riemannian manifold $M$, I prove that the $φ$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $φ$--critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^\infty(M)$--linear subspace $\sP \subset Ω^p(M…

The distance function to a generic submanifold behaves well under small perturbations.

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal su…

Rigidity for 4D Willmore submanifolds with boundary.

Morse theory connects low energy submanifolds in 3-sphere.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

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

In this paper we produce a lower bound for the number of periodic orbits of certain Hamiltonian vector fields near Bott-nondegenerate symplectic critical submanifolds. This result is then related to the problem of finding closed orbits of the motion of a charged low energy particle on a Riemannian manifold under the in…

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding …

Smooth functions on manifolds with degenerate singular submanifolds

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we…

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…

New functionals defined for free boundary minimal submanifolds in higher dimensions.

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…

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb Z_2$-taut. We explicitely construct gen…

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…

Defines a new energy for submanifolds, comparing to Willmore energy.

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

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we su…

It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold $Σ$ is contained in the volume expansion of the minimal surface which is asymptotic to $…

Variational characterization of calibrated submanifolds in different contexts.

The paper proves manifolds homeomorphic to spheres under specific Morse-Bott conditions.

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

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…

Study sharp geometric and topological properties of pinched 4D submanifolds.

We prove a formula for the normal injectivity radius(thickness)i(K,M)for C^{1,1} compact submanifolds K^k of complete Riemannian manifolds M^n in terms of geometric focal distance and double critical points. We also prove the C^1 compactness of the set of all compact submanifolds K contained in a compact subset D of a …

Decomposes smooth manifolds into algebraic submanifolds.

New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.

Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.

$ξ$-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…

Two Morse-Bott volume forms are diffeomorphic if their cohomology classes are equal.

Recent developments on biconservative submanifolds in Riemannian geometry.

Lagrangian submanifolds of a Kaehler manifold are called Hamiltonian-stationary (or $H$-stationary for short) if it is a critical point of the area functional restricted to compactly supported Hamiltonian variations. In [B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Lagrangian isometric immersions of a real-sp…

The paper studies minimal submanifolds with specific curvature properties in Euclidean space.

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q \textless{} dim M . We then construct small nearly-spherical solutio…

Let $f:M \to \mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$…

In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood…

The area renormalization procedure gives an invariant of even-dimensional closed submanifolds in a conformal manifold, which we call the Graham-Witten energy, and it is a generalization of the classical Willmore energy. In this paper, we obtain an explicit formula for the second variation of this energy at minimal subm…

We prove that the number of critical points of a Li-Tam Green's function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by con…

The paper confirms conjectures about Stein manifolds formed by quotients of the ball.

Proves properties of Morse vector fields on compact manifolds.

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.

We study the local geometry of the space of horizontal curves with endpoints freely varying in two given submanifolds $\mathcal P$ and $\mathcal Q$ of a manifold $\mathcal M$ endowed with a distribution $\mathcal D\subset T\M$. We give a different proof, that holds in a more general context, of a result by Bismut (Larg…

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 …

Study magnetic geodesics on odd spheres, computing critical energy values.

We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant metric. For…

Tight isoparametric hypersurfaces in spheres have minimal critical points.