arXiv subjects
Publications and source records attributed to Xiaodong Cao.
Let $Δ(a,b;x)$ denote the error term of the general two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value $\sum_{n\leq T}Δ^2(a,b;n)$ and the continuous mean value $\int_1^TΔ^2(a,b;x)dx$.
In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.
Suppose $a$ and $b$ are two fixed positive integers such that $(a,b)=1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $Δ_{a,b}(x)$ of the general two-dimensional divisor problem.
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.
In this paper, we prove that the first eigenvalues of $-Δ+ cR$ ($c\geq \frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.
Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds. In each case, we describe the long time behavior of the solutions of the corresponding ODE system.
In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvature.