arXiv research

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.

Researchers create higher-dimensional $\mathcal{I}^\prime$-curvatures and find counterexamples to the Hirachi conjecture.

The paper generalizes CR invariants using renormalized characteristic forms.

The study proves CR structures on specific three-manifolds are equivalent to standard structures.

The Q-curvature has been playing a central role in conformal geometry since its discovery by T. Branson. It has natural analogy in CR geometry, however, the CR Q-curvature vanishes on the boundary of a strictly pseudoconvex domain in C^{n+1} with a natural choice of contact form. This fact enables us to define a "secon…

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.

We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator…

The theory of ambient spaces is useful to define CR invariant objects, such as CR invariant powers of the sub-Laplacian, the $P$-prime operators, and $Q$-prime curvature. However in general, it is difficult to write down these objects in terms of the Tanaka-Webster connection. In this paper, we give those explicit form…

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. …

The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a co…

Totally geodesic subvarieties in moduli spaces are studied.

In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of…

Prime knots of genus one admitting diagram with at most five classical crossings were classified by Akimova and Matveev in 2014. In 2018 Kaur, Prabhakar and Vesnin introduced families of L-polynomials and F-polynomials for virtual knots which are generalizations of affine index polynomial. Here we introduce a notion of…

The paper constructs global CR invariants from renormalized characteristic forms.

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, thi…

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial st…

The main purpose of this note is the study of the total space of a holomorphic Lie algebroid $E$. The paper is structured in three parts. In the first section we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The se…

It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements …

The paper finds geodesics on specific Finsler spheres with unique properties.

Study of manifolds with prime cyclic group actions and curvature properties.

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible ho…

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples.…

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decompositio…

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formu…

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

The study finds infinitely many twist knot complements with totally geodesic surfaces.

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities.

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, a…

Totally geodesic hypersurfaces in a sphere have small total curvature.

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature,…

Paper compares total quotient curvature and proves bounds for Einstein metric.

Paper solves degenerated circle pattern metric problem in spherical geometry.

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the probl…

Ancient flows converge fast with finite curvature and convexity.

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

Unified framework for Alexandrov 3-spaces, extending manifold results.

Study on Gauss images of specific minimal surfaces with finite curvature.

In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

Paper proves total curvature for convex hypersurfaces in equiaffine space.

The study preserves upper bounds of total scalar curvature in conformal classes.

Paper proves curvature estimate for curved spaces.