arXiv research

Study of curves in dual space with constant curvature and torsion.

The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the …

Study pinched self-dual Weyl curvature in compact 4-manifolds.

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

Study curvature and torsion in Gaussian distribution's dual coordinate system.

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…

We prove that the connected sums CP_2 # CP_2 and CP_2 # CP_2 # CP_2 admit self-dual metrics with positive Ricci curvature. Moreover, every self-dual metric of positive scalar curvature on CP_2 # CP_2 is conformal to a metric with positive Ricci curvature.

Develops torsion dual connections for statistical manifolds.

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

Paper solves dual Minkowski problem in 2D plane for specific curvature cases.

Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.

This paper solves the dual Minkowski problem for q-torsional rigidity.

We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case.

Derives stability for curvature measure near constant density, proving dual Minkowski problem solutions.

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Ya…

New method solves a generalized Minkowski problem using a curvature flow.

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.

Solves a generalized dual Minkowski problem for specific values of q.

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negat…

New tensors reveal full curvature structure from Riemann tensor.

Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.

Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.

Geometric framework for inverse problems using foliations and dual connections.

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0<p<1$. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.

The paper studies singularities of pedal curves of hyperbolic frontals.

Paper solves a new Minkowski problem for a specific type of rigidity.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

In this note we prove that a (anti-)self dual quasi Yamabe soliton with positive sectional curvature is rotationally symmetric. This generalizes a recent result of G. Huang and H. Li in dimension four. Whence, (anti-) self dual gradient Yamabe solitons with positive sectional curvature is rotationally symmetric. We als…

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…

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We…

Classifies Heisenberg-invariant self-dual Einstein manifolds with explicit metrics.

We study ruled surfaces in R3 which are obtained from dual spher- ical indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean curvatures of the ruled surfaces and give some results of being Wein- garten surface.

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,…

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.

Proves conjecture about foliations on curved spaces.

In this paper, we study Mannheim surface offsets in dual space. By the aid of the E. Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic trihedron (dual Darboux frame). We obtain the relationships between the invariants …

The first aim of this paper is to define the dual timelike Mannheim partner curves in Dual Lorentzian Space D3 1, the second aim of this paper is to obtain the relationships between the curvatures and the torsions of the dual timelike Mannheim partner curves with respect to each other and the final aim of this paper is…

The first aim of this paper is to define the dual timelike - spacelike Mannheim partner curves in Dual Lorentzian Space ID3 1, the second aim of this paper is to obtain the relationships between the curvatures and the torsions of the dual timelike - spacelike Mannheim partner curves with respect to each other and the f…

We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometri…

A new formalism simplifies SO(3) Yang-Mills theory connections.

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a…

We consider self-dual metrics on 3CP^2 of positive scalar curvature admitting a non-trivial Killing field, but which is not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined b…

We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we …

We establish a $C^{1,α}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,α}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…

The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.