arXiv research

The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.

Critical spherical catenoids have Robin nullity and asymptotic radius determined.

Study on second Robin eigenvalue for Laplacian on manifolds.

The paper compares heat kernels on manifolds with Robin boundary conditions.

Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.

Paper discusses gluing formula for zeta-determinants with Robin boundary condition.

Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.

A computational technique for calculating nullity vectors and kernel vectors, using the new Finsler package, is introduced. As an application, three interesting counterexamples are given. The first counterexample shows that the two distributions $\mathrm{Ker}_R$ and $\N_R$ do not coincide. The second shows that the nul…

Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.

The study classifies Riemannian manifolds with curvature nullity.

Study on submanifolds with specific nullity ranks, proving inequalities between nullity ranks.

Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.

Upper bounds for second Robin eigenvalue on Riemannian surfaces.

Stability of Yang-Mills connections' Morse indices and nullity in 4D.

New conditions on Riemannian manifolds' curvature tensor.

The Klein-Grifone approach to global Finsler geometry is adopted. The nullity distributions of the three curvature tensors of Cartan connection are investigated. Nullity distributions concerning certain relevant special Finsler spaces are considered. Concrete examples are given whenever the situation needs.

We give a conceptual proof of the fact that if M is a complete submanifold of a space form, then the maximal integral manifolds of the nullity distribution of its second fundamental form through points of minimal index of nullity are complete.

Paper finds eigenvalue bounds for hyperbolic space domains.

Proposes a flexible tournament design combining knockout and round-robin.

For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $η$ is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity c…

The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution $\N_{R^\ast}$ is proved. Two counterexamples are given: the first shows that $\N_{R…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $ν$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two tr…

Estimates the index of the Laplace operator on planar domains with Robin boundary condition.

Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.

Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dim…

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^…

In this paper the result of real hypersurfaces in non-flat complex space forms, whose structure vector field $ξ$ belongs to the $κ$-nullity distribution is extended in case of three dimensional real hypersurfaces in non-flat complex space forms. Furthermore, generalization of notion ($κ$,$μ$)-nullity distribution defin…

Minimal surface doublings have specific index and nullity values.

In this note we investigate the $(E)$-index and the $(E)$-nullity of vacuum solutions from the two-torus into the two sphere.

Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.

Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.

Paper characterizes umbilical hypersurfaces using a generalized overdetermined problem.

The nullity of a minimal submanifold $M\subset S^{n}$ is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nulli…

We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of…

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …

The paper calculates Morse indices and nullities for embedded networks on spheres.

We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on comple…

We investigate 3-dimensional almost Kenmotsu manifolds satisfying special types of nullity conditions depending on two smooth functions $κ,μ$. When either $κ<-1$ and $μ=0$ or $h=0$, such conditions coincide with the $κ$-nullity condition which we show to be equivalent to the $η$-Einstein one. As an application of this …

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Study pseudo-laplacians and ζ(1) for spinor bundles over Riemann surfaces.

Study proves a sharp upper bound for the zero set area of a static manifold's potential.

Study calculates indices and nullities of focal manifolds in spheres.

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let $M^m$ be a complete Riemannian manifold and let $f\colon M^m\to\R^n$ be a minimal isometric immersion with index of relative nullity at least $m-2$ at any point. We show that if the Omori-Yau maximum princ…

In this paper, we consider surfaces in 4--dimensional pseudo--Riemannian space--forms with index 2. First, we obtain some of geometrical properties of such surfaces considering their relative null space. Then, we get classifications of quasi--minimal surfaces with positive relative nullity.

Paper solves capillary Lp-Minkowski problem for p>1.

Paper derives second variational formula for statistical manifold mappings.

Minimal submanifolds in spheres can be produced via Clifford type minimal products, and their Morse indices and nullities are calculated.

Study introduces a new Allen-Cahn energy on hypersurfaces and analyzes its properties.