arXiv research

Analyzes a critical spherical catenoid in hyperbolic space, proving its index and nullity.

Sturm theory applied to symplectic geometry and mechanics.

We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.

Here I show compatibility of two definition of generalized curvature bounds --- the lower bound for sectional curvature in the sense of Alexandrov and lower bound for Ricci curvature in the sense of Lott--Villani--Sturm.

Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is …

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy exampl…

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s…

Study on Sturm-Liouville problems with zero potential and Neumann boundary conditions.

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from t…

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.

Construct Lax pairs for BKM equations and related integrable hierarchies.

Study the spectrum of Page's metric on complex projective spaces.

We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov sp…

Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…

In this paper, we obtain a necessary and sufficient condition for $L^{\infty}$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{dx^2} + b(x) \frac d{dx} -V$ on an open interval of $\rr$, which is equivalent to the $L^1$-uniqueness of the associated Fokker-Planck equation. For a general elliptic operator $\LL^V:=Δ…

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

In this article, we look at the effect of volatility clustering on the risk indifference price of options described by Sircar and Sturm in their paper (Sircar, R., & Sturm, S. (2012). From smile asymptotics to market risk measures. Mathematical Finance. Advance online publication. doi:10.1111/mafi.12015). The indiffere…

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spa…

Proposes a new metric space example showing non-constant topological dimension.

Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns out that the radius of curvature and the support function of a $\mathcal{P}$-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equati…

We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $- 4 d^2/ds^2 + κ^2 (s)$ with potential given by the curvature of a closed curve.

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preservin…

Extremal spectral properties of the Lawson tori are studied. A Lawson torus carries an extremal metric for some eigenvalue of the Laplace-Beltrami operator. The main result of this paper is that the number of this eigenvalue is expressed in terms of fundamental tones of auxiliary periodic Sturm-Liouville problems.

Uniform estimates for Kaehler metrics' diameters and volumes.

In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.

It is shown that curvature-dimension bounds CD(N, k) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L^1- Poincare-inequality under some symmetry assumption on the choice of transport rays in the cut locus of (X,d). This condition is satisfied if (X,d) has m-almost surely no branching points.

The study identifies obstructions to global visibility of singularities in spacetimes.

The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension c…

We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the cur…

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…

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Mor…

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang-Schroeder-Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. W…

In this paper we prove first order differential Harnack estimates for positive solutions of the heat equation (in the sense of distributions) under closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the Chern connection, $\mathbf{S}-$curvature and Hessian) and suitable Ricci curvature bounds throug…

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, …

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is known that the $ζ$-function of this operator $ζ_L(s)=\sum_{λ\in\spec(L)\setminus\{0…

New estimates for Green's functions in varying Kähler metrics.

We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for the configuration space a lower Ricci curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality, gradient estimates and Wa…

This paper improves Green's function estimates for compact Kähler manifolds.

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the stat…

It was proved by H. Chen earlier that the property of the sum of any two eigenvalues of the curvature operator is positive is preserved under the ricci flow in all dimensional. By a recent result of Phong-Sturm, a similar notion of positive 2-traceless bisectional curvature positive is preserved on complex surface. We …

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Son…

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems involving the support function. We give a practical application by provin…

We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähle…