On the cross curvature flow

Abstract

In this paper, we study the cross curvature soliton. We study the cross curvature soliton with a warped product structure. On the other hand, we show that the volume entropy is decreasing along the cross curvature flow.

Introduction

We first recall the definition of the cross curvature. Suppose $(M,g)$ is a 3-dimensional manifold. We define the tensor

$$P_{ij}=R_{ij}-\frac{1}{2}R{g}_{ij},$$

where $R_{ij}$ is the Ricci curvature tensor and R is the scalar curvature. We can raise the indices as follows:

$$P^{ij}=g^{ik}{g}^{jl}{R}_{kl}-\frac{1}{2}R{g}^{ij}.$$

When $P^{ij}$ has an inverse $V_{ij}$, the cross curvature tensor is defined as

$$h_{ij}=(\det P)V_{ij},$$

where

$$\det P=\left(\frac{\det P^{kl}}{\det g^{kl}}\right).$$

Since the tensor $P_{ij}$ is symmetric, we can diagonalize it at any given point $p\in M$. That is, we can choose an orthonormal basis at $p\in M$ such that $P_{ij}$ is diagonal and $g_{ij}={\delta }_{ij}$. Hence, the Ricci curvature tensor $R_{ij}$ and the cross curvature tensor $h_{ij}$ are also diagonal. In particular, if the eigenvalues of $P_{ij}$ are $a=-R_{2323}$, $b=-R_{1313}$, $c=-R_{1212}$, then the eigenvalues of $R_{ij}$ are $-(b+c)$, $-(a+c)$, $-(a+b)$ and the eigenvalues of $h_{ij}$ are bc, ac, ab. Hence, if $(M,g)$ has negative sectional curvature, then both $P_{ij}$ and $h_{ij}$ are positive definite.

Throughout this paper, we assume that $(M,g_0)$ is a 3-dimensional manifold with negative sectional curvature. The cross curvature flow is defined as the evolution equation of the metric $g=g(t)$ satisfying

$$\frac{\partial }{\partial t}{g}_{ij}=2h_{ij}$$

with the initial condition $g{|}_{t=0}=g_0$. This was first introduced and studied by Chow and Hamilton in [12]. The short time existence of the cross curvature flow was proved by Buckland in [3]. Backward uniqueness of the cross curvature flow was obtained by Kotschwar in [23]. It was expected that the cross curvature flow exists for all time and converges to a limiting metric with constant negative sectional curvature. See [13], [22] for results in this direction. See also [8], [9], [10], [25] and the references therein for more information about cross curvature flow.

As the long time existence and the convergence of the cross curvature flow are still not known in general, we study in this paper the cross curvature soliton, which is a special kind of solution to the cross curvature flow. More precisely, the cross curvature soliton is the self-similar solution of the cross curvature flow and has been introduced in [18]: Given a 3-dimensional manifold $(M,g_0)$ with negative sectional curvature, $g=g(t)$ is a cross curvature soliton if

$$g(t)=\sigma (t){\varphi }_t^{⁎}(g_0)$$

where $\sigma :[0,T)\to (0,\infty )$ is a smooth function with $\sigma (0)=1$, and ${\varphi }_t:M\to M$ is a one-parameter family of diffeomorphisms with ${\varphi }_0=i{d}_M$ being the identity map on M. The following theorem was proved in [11, Lemma B.27].

Theorem 1.1

Any compact cross curvature soliton (1.6) must have constant negative sectional curvature.

In view of Theorem 1.1, we consider the noncompact cross curvature solution. In section 2, we study the cross curvature soliton from the equation point of view. In section 3, we study the cross curvature soliton with warped product structure. In particular, we are able to find nontrivial cross curvature solitons on a noncompact manifold. See Example 3.4 and Example 3.5. We remark that, while the Ricci soliton, which is the self-similar solution of the Ricci flow, has been studied extensively (see [1], [2], [4], [5], [6], [7], [16], [28], [31], [33] and the references therein), there are not many results for the cross curvature soliton in the literature. So our paper can be viewed as the first step to understand the cross curvature soliton.

In another direction, we study the volume entropy along the cross curvature flow. Given a compact Riemannian manifold $(M,g)$, the volume entropy is defined by

$$v(g)=\underset{r\to \infty }{\lim \inf }\frac{\log vol(B_g(\tilde{x},r))}{r},$$

where $\tilde{x}$ is any point of the universal cover $\tilde{M}$ of M and $B_g(\tilde{x},r)$ is the ball of radius r centered at $\tilde{x}$ in M for the metric lifted from g. In [26], Manning considered the volume entropy along the Ricci flow on a surface. In particular, he showed that the volume entropy is decreasing along the normalized Ricci flow on a compact surface of negative Euler characteristic. In [20], Jane constructed examples of smooth metrics on compact surface of nonnegative Euler characteristic such that the volume entropy is increasing along the normalized Ricci flow. One wonders if a similar result holds in higher dimension. In [30], Suàrez-Serrato and Tapie studied the volume entropy along the Yamabe flow. See also [32].

Inspired by these results, we prove in section 4 the following:

Theorem 1.2

Let $(M,g_0)$ be a 3-dimensional manifold with negative sectional curvature. Suppose that $g=g(t)$ is the solution of the cross curvature flow (1.5) defined on $[0,\infty )$, with initial condition $g{|}_{t=0}=g_0$. Then the volume entropy of $g(t)$ is decreasing in t.

As mentioned above, it was expected that the cross curvature flow exists for all time and converges to a limiting metric with constant negative sectional curvature. If this is true, Theorem 1.2 then implies the following entropy-rigidity: Among all the metrics with negative sectional curvature, the constant negative sectional curvature metric has the minimum volume entropy.