arXiv research

Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.

Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume…

The study shows ends of shrinking gradient $ρ$-Einstein solitons are non-parabolic.

In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball $\mathbf{D}^3$, which contains multiple horizons.

The Nirenberg problem yields multiple conformal metrics for a given scalar curvature function.

Maps are shown to be Riemannian products with Ricci-flat fibers.

Study constructs $H$-space on metric spaces, providing rigidity criteria.

Study finds multiple periodic solutions to ODEs related to curvature problems.

A new method detects hidden driving forces in systems with multiple observables.

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a We…

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.

A new framework for hyperbolic neural networks using the Klein model is introduced.

The paper proves solutions for Yamabe equations on manifolds with boundary.

Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.

We study multiplicity of constant scalar curvature metrics in products of a compact closed manifold and a compact manifold with boundary using equivariant bifurcation theory.

Gradient almost para-Ricci-like solitons have constant coefficients and scalar curvatures.

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain co…

We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore …

We show that, for each alpha in the interval (-1,1), the only Riemannian metrics on the space of positive definite matrices for which the alpha and -alpha-connections are mutually dual are matrix multiples fo the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to s…

In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Mo…

Neural network predicts functional responses from scalar inputs.

A classic result by Gromov and Lawson states that a Riemannian metric of non--negative scalar curvature on the Torus must be flat. The analogous rigidity result for the standard sphere was shown by Llarull. Later Goette and Semmelmann generalized it to locally symmetric spaces of compact type and nontrivial Euler chara…

Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.

Paper proposes new costs for learning multiple centers in MDNs.

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (o…

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a riemannian compact $4-$dimensional manifold $(M^4,g_0)$ with positive scalar curvature. We prove new exitence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri-Coron and…

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^…

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold…

Tree-based algorithm for functional data analysis reduces generalization error.

Study on positive solutions of Yamabe-type equation on spheres.

We showed the existence of non-radial solutions of the equation $Δu -λu + λu^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $λ$. We show that for any isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is a regular level set (and the number …

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on …

Study of para-Ricci-like solitons on specific Riemannian manifolds.

For simple Lie groups, the only homogeneous manifolds $G/K$, where $K$ is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the c…

New mappings in Minkowski spacetime classified under mild conditions.

This paper explores conditions for positive scalar curvature on spin^c manifolds.

Motivated by AdS/CFT, the extension is made to spin-half of a scalar calculation of the conformal anomalies and functional determinants of GJMS operators. The formal aspects are heuristic but sufficient. A Barnes zeta function representation again proves effective. The determinants are calculated for the two factorisat…

The clustering algorithms that view each object data as a single sample drawn from a certain distribution, Gaussian distribution, for example, has been a hot topic for decades. Many clustering algorithms: such as k-means and spectral clustering are proposed based on the single sample assumption. However, in real life, …

The paper proves curvature inequalities for submanifolds in space forms.

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero b…

Proof confirms cosmic censorship for charged gravitational collapse.

This work overcomes bias in concave multi-objective reinforcement learning.

This paper analyzes MORL and proposes efficient algorithms to learn Pareto optimal policies.

New mechanisms improve differential privacy for scalar queries.

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + δh)$ has at least $Cat(M) +1$ solutions for $δ$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-categ…

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…