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Gunhee Cho

Publications and source records attributed to Gunhee Cho.

32 records · Page 2Linked to original sources
arXiv2023

The Stochastic Schwarz lemma on Kähler Manifolds by Couplings and Its Applications

We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall-Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete non-compact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a byproduct, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

math.DG↗
arXiv2023

Non-existence of complete Kähler metric of negatively pinched holomorphic sectional curvature

We show the theorem which provides some sufficient condition to the non-existence of a complete Kähler--Einstein metric of negative scalar curvature whose holomorphic sectional curvature is negatively pinched: Let $Ω$ be a bounded weakly pseudoconvex domain in $\mathbb{C}^n$ with a Kähler metric $ω$ whose holomorphic sectional curvature is negative near the topological boundary of $Ω$ (with respect to relative topology of $\mathbb{C}^n$) and $ω$ admits the quasi-bounded geometry. Then $ω$ is uniformly equivalent to the Kobayashi--Royden metric and the following dichotomy holds: 1. $ω$ is complete, and $ω$ is uniformly equivalent to the complete Kähler--Einstein metric with negative scalar curvature. 2. $ω$ is incomplete, and there is no complete Kähler metric with negatively pinched holomorphic sectional curvature. Moreover, $Ω$ is Carathéodory incomplete. Our approach is based on the construction of a Kähler metric of negatively pinched holomorphic sectional curvature and applying the implication of equivalence of invariant metrics inspired by Wu-Yau.

math.DG↗
arXiv2023

Sharp weighted CR trace Sobolev inequalities

We establish a sharp Sobolev trace inequality on the Siegel domain $Ω_{n+1}$ involving the weighted norm-$W^{2,2}(Ω_{n+1}, ρ^{1-2[γ]})$. The inequality is closely related the realization of fractional powers of the sub-Laplacian on the Heisenberg group $H^n=\partial Ω_{n+1}$ as generalized Dirichlet-to-Neumann operators associated to the weighted poly-sublaplacian, generalizing observations of Frank--González--Monticelli--Tan.

math.AP↗
arXiv2022

Vanishing results from Lichnerowicz Laplacian on complete Kähler manifolds and applications

In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact Kähler manifolds.

math.DG↗
arXiv2022

Sub-Riemannian Geodesics on $SL(2, \mathbb{R})$

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.

math.DG↗
arXiv2021

Octonionic Brownian Windings

We define and study the windings along Brownian paths in the octonionic Euclidean, projective and hyperbolic spaces which are isometric to 8-dimensional Riemannian model spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior.

math.PR↗
arXiv2021

Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature bound

We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.

math.DG↗
arXiv2021

The Subelliptic Heat Kernel of the Octonionic Anti-De Sitter Fibration

In this note, we study the sub-Laplacian of the 15-dimensional octonionic anti-de Sitter space which is obtained by lifting with respect to the anti-de Sitter fibration the Laplacian of the octonionic hyperbolic space $\mathbb{O}H^1$. We also obtain two integral representations for the corresponding subelliptic heat kernel.

math.DG↗
arXiv2020

The subelliptic heat kernel of the octonionic Hopf fibration

We study the sub-Laplacian of the $15$-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics of the heat kernel and explicitly compute the sub-Riemannian distance.

math.DG↗
arXiv2019

The Kobayashi-Royden metric on punctured spheres

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere $\mathbb{CP}^1\backslash\{0,1,\infty\}$ in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard's theorem as an application of the asymptotic expansion. Meanwhile, the approach in the paper leads to the conclusion that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit metric formula and the conclusion regarding the coefficients apply to a more general case as well, the metric on $\mathbb{CP}^1\backslash\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}$ will be given as a concrete example of our results.

math.DG↗
arXiv2019

Invariant metrics on the Complex ellipsoid

We provide a class of geometric convex domains on which the Carathéodory-Reiffen metric, the Bergman metric, the complete Kähler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Poincaré-disk.

math.MG↗