CR流形的幾何型發展方程問題之研究

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2009 Research Report

Chin-Tung Wu

I work in the area of geometric evolution equations on CR manifolds. During the past year, August 2008 - July 2009, my research concentrated on the entropy formulas for the CR heat equation and its applications, CR Yamabe flow and the CR analogue of Liouville-type theorem.

Conference and Workshop:

1. _{“Eigenvalues and Entropy formula of Geometric flows on CR 3-manifolds”,}

Nonlinear Analysis and Geometry Analysis, Chi-tou, Sep. 5 - 8, 2008.

2. _{The Ninth Pacific Rim Geometry Conference, Taiwan University, Taipei, Dec. 10}

- 14, 2008.

3. _{“The entropy formulas for the CR heat equation and its applications”, AMMS}

2008, Tsing-Hua University, Hsinchu, Dec. 19 - 21, 2008.

4. _{“The entropy formulas for the CR heat equation and its applications”, Sun}

Yat-Sen University, Guangzhou, China, Jan. 8, 2009.

5. _{“The CR Harnack Estimates and their applications ”, East China Normal}

University, Shanghai, China,March 31, 2009 .

6. _{“The CR Volume Comparison ”, Taiwan University, Taipei, April 14, 2009.}

Advance in research:

1. _{We derive Perelman's and Nash-type entropy formulas for the CR heat equation}

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sublaplacian on a closed pseudohermitian (2n+1)-manifold. (Joint with Shu-Cheng Chang).

2. _{We obtain a CR analogue of Li-Yau-Hamilton inequality for the Yamabe flow on}

a closed spherical CR 3-manifold with positive Yamabe constant and vanishing torsion. By combining this parabolic subgradient estimate with a compactness theorem of contact classes, it follows that the CR Yamabe flow exists for all time and converges smoothly to, up to the CR automorphism, a unique limit contact form of positive constant Tanaka-Webster scalar curvature. As a consequence, there is a contact form of positive constant Tanaka-Webster curvature on a closed spherical CR 3-manifold with positive Yamabe constant and vanishing torsion (Joint with Shu-Cheng Chang).

3. _{We show the natural CR analogue of Liouville-type theorem holds for the}

positive pseudoharmonic function in a complete pseudohermitian 3-manifold with vanishing torsion and nonnegative Tanaka-Webster scalar curvature. As a consequence, we obtain the global subgradient estimate as well. In particular, we recapture Koranyi and Stanton's result for Liouville-type theorem where their method is only worked on the Heisenberg group. Finally we present an alternating proof of Liouville-type theorems via the CR heat equation (Joint with Shu-Cheng Chang).

Plans to study:

1. _{We will study the Cartan flow on a closed CR 3-manifold with vanishing torsion.} 2. _{We will study some properties about the CR Obata’s Theorem without the}

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