arXiv research

Develops tensor calculus for submanifolds of arbitrary codimension.

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

Novel coarse extrinsic curvature for Riemannian submanifolds.

Derivative formulas on measure spaces of Riemannian manifolds are characterized.

For a compact spin manifold $M$ isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues.

New operators and curvatures derived from embedded manifolds.

Study on biharmonic map heat flow with monotonicity formula.

New CRB derived for curved models using extrinsic geometry.

Researchers define residue families and use them to solve singular Yamabe problems.

We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…

The paper derives inequalities for Riemannian submersions and applies them to specific space forms.

We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T…

Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…

We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,α}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)< α\leq 1$, then $y(M)$ is a surface of bounded extrinsic curvature. This is proved by showi…

Explicit formulas for extrinsic Paneitz operators and Q-curvatures for totally umbilic hypersurfaces.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

The study defines and characterizes extrinsic catenaries in hyperbolic space.

Study submanifolds with relative nullity in space forms using splitting tensor.

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.

Derives GJMS operators and Q-curvatures for submanifolds.

Flat tori in 3-sphere have π extrinsic diameter under specific conditions.

Uniqueness of 1D bi-Schrödinger flow proven from flat torus to compact space.

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

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

Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.

Stability of a new map derived from the equator map is analyzed.

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.

We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situatio…

Analyzes conformal anomaly in five dimensions, identifying new boundary conformal invariants.

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones.

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…

Classifies minimal submanifolds in complex hyperbolic spaces.

In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…

We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.

Proves uniqueness of geometric flow in various Riemannian manifolds.

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on t…

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the subm…