A Gradient Estimate for the Ricci–Kähler Flow

We show that for a complete solution to theRicci–Kähler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.     

Geometry of Ricci Solitons

Ricci solitons are natural generalizations of Einstein metrics on one hand, and are special solutions of the Ricci flow of Hamilton on the other hand. In this paper we survey some of the recent developments on Ricci solitons and the role they play in the singularity study of the Ricci flow.     

Uniform Sobolev Inequality along the Sasaki–Ricci Flow

In this paper we prove a uniform Sobolev inequality along the Sasaki–Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.     

Gradient Flow of the Lβ-Functional

In this paper,we start to study the gradient flow of the functional L_β introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a K?hler surface,then the symplectic property is preserved along the gradient flow.Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form.When β=1,we derive a monotonicity formula for the flow.As applications,we show that the l-tangent cone of the flow consists of the finite flat planes.     

Existence of the Self-Similar Solutions in the Heat Flow of Harmonic Maps

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On the Existence of Harmonic Solutions of the Forced Lienard Equation

In this paper, we present a method to study the existence of the harmonicsolutions of the forcde Lienard equation′s equivalent system     

An Almost Complex Chern–Ricci Flow

We consider the evolution of an almost Hermitian metric by the (1, 1) part of its Chern–Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern–Ricci flow if the complex structure is integrable and with the Kähler–Ricci flow if moreover the initial metric is Kähler. We find the maximal existence time for the flow in term of the initial data and also give some convergence results. As an example, we study this flow on the (locally) homogeneous manifolds in more detail.     

On the Approximation of Continuous Function by Harmonic Means

Let f(x) be a continuous and periodic function with period 2π and [f]=a_o/2+sum from n=1 to ∞(a_ncos nx+b_nsin nx) be its Fourier series. Denote by s_n(f,x) the n-th partial sums of [f] and by ω(f,δ) the moduls of continuity of f(x). When ω(δ) is a modul of continuity, we denote by H[ω]。the class of all functions for which ω(f,t)≤ω(t). It is well known that the n-th harmonic means and the n-th Cesro means of f∈C_2, are defined as##属性不符     

On the Geometry of Horizontally Homotheitic Maps and Harmonic Morphisms

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On the Regularity for Stationary Harmonic Maps

We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.     

A Scalar Curvature Bound Along the Conical Kähler–Ricci Flow

Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler–Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.     

Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Émery–Ricci curvature if the space of weighted L ² harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.     

The Isoperimetric Inequality in Steady Ricci Solitons

The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over S^n-1is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric,the author utilize the result of Guan-Li-Wang to get his conclusion. For the sake of the soliton structure, the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally s...     

Solitons in the system of coupled Hirota–Maxwell–Bloch equations

We propose the system of coupled Hirota–Maxwell–Bloch equations which governs the propagation of optical pulses in an erbium doped nonlinear fibre with higher order dispersion, self-steepening and self induced transparency (SIT) effects. The Lax pair is explicitly constructed and the soliton solution is obtained using the Darboux–Bäcklund transformations. Hence, the system is found to admit soliton type lossless wave propagation.     

Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

On the Kahler-Ricci Flow on Projective Manifolds of General Type

This note concerns the global existence and convergence of the solution for Kahler-Ricci flow equation when the canonical class, Kx, is numerically effective and big. We clarify some known results regarding this flow on projective manifolds of general type and also show some new observations and refined results.     

On the Curvature of Kähler Manifolds with Zero Ricci Tensor

Mathematical Notes - The behavior of the modulus of the curvature tensor and of the holomorphic sectional curvature on Ricci-flat Kähler manifolds is investigated.