arXiv research

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

Characterizes stable sheaves for equality in orbifold BG inequality.

Study proves rigidity for Heintze-Karcher inequality in substatic manifolds.

Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.

Characterizes when the Roller boundary equals the Poisson boundary of CAT(0) cube complexes.

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iter…

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

Let M be a bounded open plane domain. Let f be a continuous function on the closure of M, 3-times continuously differentiable in M, which vanish on the boundary. Polterovich and Sodin proved that the values of f cannot exceed the norm of the hessian of f, averaged over the entire domain M. In this paper we study the eq…

We relate generalized Lebesgue decompositions of measures in terms of curve fragments (Alberti representations) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space $(X,μ)$: the local norm of a form $df$ sees how fas…

Proposes ENVAR for causal discovery in structural VAR models with equal noise variance.

This work characterizes asymptotically hyperbolic 3-manifolds via CMC-foliations.

We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk …

The study provides foundations for naive diversification, a preference for equal treatment of alternatives.

The paper tackles fair classification with multiple sensitive features.

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal t…

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

Article revisits and solves Alexandrov-Fenchel inequalities for compact Kähler manifolds.

The article characterizes complex torus quotients with numerical conditions.

Study on knot properties, showing relation between unknotting and crossing numbers.

Study shows spheres in high dimensions have maximum volume if they are smooth and have a specific reach.

A new tail-shape index based on Value at Risk and Expected Shortfall.

The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then ${\rm Cap}(K)\geq (n-1)\,H_0{\rm vol}(\partial K)$. When $K$ is contai…

Dropout biases neural networks by equalizing hidden node weights.

We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be…

Study on charged parallel spinors and mass-charge inequalities.

In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, Pólya-Szegö and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is char…

Characterizes closures of test configurations and algebraic singularity types.

Algorithm ensures fairness in online classification with partial feedback.

Characterizes adequate links using Jones polynomial and crossing number.

We show that an embedded minimal annulus $Σ^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique…

Characterizes Euclidean balls with lower bounded k-th mean curvature.

New research shows unlabeled data is equally valuable as labeled data in certain semi-supervised learning scenarios.

To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…

Cylinders in warped product spaces have zero curvature.

A new bias score method optimizes fairness in classification.

In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its fl…

New link topology connects permutation discrepancies to Diaconis-Graham inequalities.

This paper characterizes Milnor invariants using diagrammatic methods.

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1)$. It is well known that the bottom of spectrum $λ_{0}$ of its unverversal covering satisfies $λ_{0}\leq(n-1) ^{2}/4$. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

The paper refines the three-page index for links, proving a new bound and characterizing specific links.

We give a geometric characterization of compact Riemann surfaces admitting orientation reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non-empty real part. We show that there is a family of disjoint simple closed geodesics th…

SCS identifies a range of plausible equally weighted portfolios, quantifying selection uncertainty.

In this paper, we prove the Hijazi inequality on compact Riemannian spin manifolds under two boundary conditions: the condition associated with a chirality operator and the Riemannian version of the $\MIT$ bag condition. We then show that the limiting-case is characterized as being a half-sphere for the first condition…

For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are charac…

The paper proves a theorem and characterizes connections over normal varieties.

Study on warped product submanifolds in Kenmotsu manifolds.