arXiv research

Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.

Local and global classifications of Einstein submanifolds in Euclidean space.

Proves a special type of submanifolds in a curved space.

The study of real Einstein submanifolds in Kähler geometry.

The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.

Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.

Develops methods for computing conformal invariants of submanifolds.

Kähler-Einstein metrics on toric submanifolds cannot be induced by projective immersions.

Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.

Study rigidifies Einstein manifolds with symmetry, proving conjecture.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

Derives GJMS operators and Q-curvatures for submanifolds.

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic…

H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro for isometric immersions of Riemannian manifolds with constant sectional curvature.

Lying at the intersection of Ado's theorem and the Nash embedding theorem, we consider the problem of finding faithful representations of Lie groups which are simultaneously isometric embeddings. Such special maps are found for a certain class of solvable Lie groups which includes all Einstein and Ricci soliton solvman…

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almos…

Solves classical problem with Kähler-Einstein metrics in complex projective spaces.

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian…

We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ h…

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

Study on stability of minimal submanifolds in specific Einstein manifolds.

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

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…

Given a minimal Lagrangian submanifold L in a negative Kaehler--Einstein manifold M, we show that any small Kaehler--Einstein perturbation of M induces a deformation of L which is minimal Lagrangian with respect to the new structure. This provides a new source of examples of minimal Lagrangians. More generally, the sam…

The paper studies Einstein submanifolds in Euclidean space and finds conditions for their properties.

A well-known result asserts that any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic. In this note, we show that this conclusion remains valid for the larger class of Einstein manifolds. As an application, when assu…

In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curva…

We consider $F: M \to N$ a minimal oriented compact real 2n-submanifold M, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, and scalar curvature R. We assume that $n \geq 2$ and F has equal Kaehler angles. Our main result is to prove that, if n = 2 and $R \neq 0$, then F is either a complex submanif…

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply …

This article describes some geometric invariants and conformal anomalies for conformally compact Einstein manifolds and their minimal submanifolds which have recently been discovered via the Anti-de Sitter/Conformal Field Theory correspondence.

Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.

Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayle…

For a given minimal Legendrian submanifold $L$ of a Sasaki-Einstein manifold we construct two families of eigenfunctions of the Laplacian of $L$ and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the case the lower bound is attained, we prove that $L$ is totally geodesic and a rig…

The paper explores p-biharmonic hypersurfaces in Einstein and conformally flat spaces.

In this paper, we give some convergence results of Lagrangian mean curvature flow under some stability conditions in a general Kähler-Einstein manifold. In particular, we prove that the flow will converge if the initial data is some small perturbation of stable minimal Lagrangian submanifold in a Kähler-Einstein manifo…

Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.

We define broadly-pluriminimal immersed 2n-submanifold F: M --> N into a Kaehler-Einstein manifold of complex dimension 2n and scalar curvature R. We prove that, if M is compact, n \geq 2, and R < 0, then: (i) Either F has complex or Lagrangian directions; (ii) If n = 2, M is oriented, and F has no complex directions, …

In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagra…

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on d…

Starting with a compact hyperbolic cone-manifold of dimension greater than or equal to 3, we study the deformations of the metric with the aim of getting Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial i…

This paper is a continuation of a paper with the same title of the last two authors. In the first part of the present paper, we give a unified geometric proof that both focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. In the second part, we complete…

A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…

`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing tec…

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 $…

The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein ([TY1], [QTY]). Inspired by A.Gray's view, the present paper shows that, these focal sets are all A- manifolds but rarely Ricci parallel, except possibly for the only unclassified cas…

The paper examines conditions for Lagrangian surfaces in Kähler-Einstein manifolds.

We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a Lagrangian submanifold of N, or have constant Kaehler angle, depending on n=1, n=2, or …