arXiv research

The paper defines Hesse solitons and explores their properties on Hessian manifolds.

We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results g…

The study explores Hesse manifolds and their symmetries in multifield cosmological models.

The paper explores polynomial functions with bounded Hess^+ complements and their properties.

Automorphisms of Hessenberg varieties are algebraic tori of dimension n-1.

Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.

Let $P_t$ be the diffusion semigroup generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(σ^2ρ_o^2+c)$ for some constants $σ, c>0$ and $ρ_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the…

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. A…

The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.

New metrics and coordinates for barcode space using group theory.

We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = ψ(x)$ where $f$ is a "natural" differential operator, with a restricted domain $F$, on a manifold $X$. By "natural" we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be c…

Study heat flow inequalities on 1-forms in RCD spaces.

We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomo…

Generalizes Li-Yau Harnack inequality to path space of manifolds.

Optimizes shapes on non-standard manifolds.

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic …

Let $M$ be a compact Riemannian manifold and $h$ a smooth function on $M$. Let $ρ^h(x)=\inf_{|v|=1}\left(Ric_x(v,v)-2Hess(h)_x(v,v) \right)$. Here $Ric_x$ denotes the Ricci curvature at $x$ and $Hess(h)$ is the Hessian of $h$. Then $M$ has finite fundamental group if $Δ^h-ρ^h<0$. Here $Δ^h=: Δ+2L_{\nabla h}$ is the Bis…

We explain how the Harish-Chandra Plancherel Theorem and results in relative Lie algebra cohomology can be used in order to compute in a uniform way the $L^2$-Betti numbers, the Novikov-Shubin invariants, and the $L^2$-torsion of compact locally symmetric spaces thus completing results previously obtained by Borel, Lot…

The aim of this paper is to classify three dimensional compact Riemannian manifolds $(M^{3},g)$ that admits a non-constant solution to the equation $-Δf g+Hess f-fRic=μRic+λg,$ for some special constants $(μ, λ)$, under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will…

We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families a…

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. I…

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated …

Proves weakly non-collapsed RCD spaces are strongly non-collapsed.

In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of L…

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $\left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert Δu\right\Vert _{L^p},$ valid a priori for all smooth functions $…

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $2π\int_{θ\in S^1}χ(Σ_θ)\geq \frac{1}{2}\int_{θ\in S^1}\int_{Σ_θ}(|du|^{-2}|Hess(u)|^2+R_M)$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Σ_θ=u^{-1}\{θ\}$. As our prima…

New superintegrable systems derived from Frobenius structures.

Consider a smooth manifold $M$. Let $G$ be a compact Lie group which acts on $M$ with cohomogeneity one. Let $Q$ be a singular orbit for this action. We study the gradient Ricci soliton equation $\Hess(u)+\Ric(g)+\fracε{2}g=0$ around $Q$. We show that there always exists a solution on a tubular neighbourhood of $Q$ for…

Survey of global Calderón-Zygmund inequalities on Riemannian manifolds.

New example shows non-compact manifolds can lack $L^p$-Calderón-Zygmund inequalities.

Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex co…

Study of Disc-structure space of compact smooth manifolds.

Graph Laplacians converge under symmetric divergence conditions.

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$ and 0 is a regular value of…

Study classifies 4D shrinkers with nonnegative Ricci curvature.

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniquenes…

Enhances SMC² with Hessian info for more efficient posterior approximation.

The study describes special real manifolds and invariant admissible cubics in Vinberg cones.

New flows introduced for symplectic geometry.

The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.

Proves uniqueness of geometric flow in various Riemannian manifolds.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

Investigate scalar curvature under geometric flows

Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…

The article calculates the $\mathbb{F}$-convergence rate for Ricci flows with closed and smooth tangent flows.

Paper introduces Tensor Gauge Flow Models for better data encoding.