# Monotonicity of functionals along conformal Ricci flow

@article{Li2020MonotonicityOF, title={Monotonicity of functionals along conformal Ricci flow}, author={Fengjiang Li and Peng Lu and Jian-hong Wang and Yu Zheng}, journal={Proceedings of the American Mathematical Society}, year={2020} }

The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.

## References

SHOWING 1-10 OF 15 REFERENCES

### A note on conformal Ricci flow

- Mathematics
- 2011

In this note we study the conformal Ricci flow that Arthur Fischer introduced in 2004. We use DeTurck's trick to rewrite the conformal Ricci flow as a strong parabolic-elliptic partial differential…

### Evolution of an extended Ricci flow system

- Mathematics
- 2008

has been used with great success for the construction of canonical metrics on Riemannian manifolds of low dimension. In his first paper on the Ricci flow, Hamilton proved that given an initial metric…

### Conformal Ricci flow on asymptotically hyperbolic manifolds

- MathematicsScience China Mathematics
- 2018

In this article, we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi’s type curvature derivative estimate for conformal Ricci…

### The entropy formula for the Ricci flow and its geometric applications

- Mathematics
- 2002

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…

### Hamilton's Ricci Flow

- Mathematics
- 2018

Riemannian geometry Fundamentals of the Ricci flow equation Closed 3-manifolds with positive Ricci curvature Ricci solitons and special solutions Isoperimetric estimates and no local collapsing…

### Asymptotic-behavior for singularities of the mean-curvature flow

- Mathematics
- 1990

is satisfied. Here H(p,ή is the mean curvature vector of the hypersurface Mt at F(/?, t). We saw in [7] that (1) is a quasilinear parabolic system with a smooth solution at least on some short time…

### An introduction to conformal Ricci flow

- Mathematics
- 2003

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the…

### ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS

- Mathematics
- 2013

HONGXIN GUO, ROBERT PHILIPOWSKI, AND ANTON THALMAIERAbstract. In this note we determine the ﬁrst two derivatives of the clas-sical Boltzmann-Shannon entropy of the conjugate heat equation on…

### Ricci flow coupled with harmonic map flow

- Mathematics
- 2009

We investigate a new geometric flow which consists of a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map phi from M to some closed target manifold N with a…