arXiv research

Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.

The paper verifies a conjecture about the index of symmetric spaces.

Classifies totally geodesic submanifolds in exceptional symmetric spaces.

New tools compute index of symmetry in homogeneous fibrations.

Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less or equal …

This note finds the index for specific types of symmetric spaces.

Researchers solved a 30-year-old conjecture about exceptional symmetric spaces.

We find the index of $\widetilde{\nabla}$-quasi-conformally symmetric and $\widetilde{\nabla}$-concircularly symmetric semi-Riemannian manifolds, where $\widetilde{\nabla}$ is metric connection.

Study estimates index of minimal hypersurfaces using Betti numbers.

Finite index solutions to Bernoulli problem are always axially symmetric.

Study finds a catenoid's Morse index is 4.

Study finds minimal hypersurfaces in compact symmetric spaces have a linear index bound.

Compact holonomy groups found in symmetric spaces.

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

The paper studies symmetrization effects on Lelong numbers and Monge-Ampère masses of plurisubharmonic functions.

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equ…

Formula for Z_2-valued index of symmetric operators on manifolds.

Study introduces index for mean curvature flow shrinkers and proves gap theorem.

Study examines kurtosis in heavy-tailed symmetric stable distributions.

Upper bound on index of rotationally symmetric self-shrinking tori.

In this short article, we find an explicit formula for Maslov index of Whitney n-gons joining intersections points of n half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an n-gon with the fat diagonal in the symmetric product.

The study examines the index of MOTS in Kerr-Newman-de Sitter spacetime and its relation to mass and charge.

Formula for twisted orbital integrals using hypoelliptic Laplacian.

New minimal surface theory disproves a conjecture in symmetric spaces.

The study finds robust index bounds for minimal hypersurfaces in specific geometric spaces.

Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.

Paper computes G-index for specific hyperbolic manifolds.

Higher index theorem for Dirac operators on finite-volume spaces.

Study Morse complexity of manifolds and homology classes, proving bounds and implications.

New symmetric Willmore tori emerge from Clifford torus in Berger spheres.

We prove an extension of the Index Theorem for Morse-Sturm systems of the form $-V''+RV=0$, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in…

Deep single-index Fréchet regression for metric space-valued outputs

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate po…

Study of spectral flow in symmetric Toeplitz operator families.

We study the index of symmetry of a compact generalized flag manifold M=G/H endowed with an invariant Kaehler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces and we estimate their dimension.

Paper calculates Morse index of Y-singular minimal surfaces.

New spaces identified with specific properties.

We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.

New risk index measures insolvency risk using fractal geometry of balance sheets.

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct ma…

The holonomy algebra of a pseudo-hyper-Kählerian manifold of signature $(4,4n+4)$ is a subalgebra of $\sp(1,n+1)$. Possible holonomy algebras of these manifolds are classified. Using this, a new proof of the classification of simply connected pseudo-hyper-Kählerian symmetric spaces of index 4 is obtained.

Abstract Morse index theorem applied to various optimization problems.

It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincaré-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$ is non-positive. Conversely, any non-positive half-integer arises in this fashion. …

We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the fam…

Study f-minimal hypersurfaces in weighted manifolds, finding index bounds.

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…

We give a formula for the parity of the Maslov index of a triple of Lagrangian subspaces of a skew symmetric bilinear form over the real numbers. We define an index two subcategory (the even subcategory) of a 3-dimensional cobordism category. The objects of the category are surfaces are equipped with Lagrangian subspac…