CR Li-Yau Gradient Estimate and Perelman Entropy Formulae Shu-Cheng Chang National Taiwan University The 10th Paci…c Rim Geometry Conference Osaka-Fukuoka, Part I, Dec. 1-5, 2011 The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Contents Motivations Pseudohermitian 3 -Manifold The CR Li-Yau Gradient Estimate The CR Li-Yau-Hamilton and Li-Yau-Perelman Harnack Estimate Perelman Entropy Formulas and Li-Yau-Perelman Reduce Distance Th e Proofs The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Motivations Problem geometrization problem of contact 3-manifolds via CR curvature ‡ows The Cartan Flow : Spherical CR structure The torsion ‡ow : the CR analogue of the Ricci ‡ow The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
The Torsion Flow The torsion ‡ow � ∂ t J ( t ) = 2 A J , θ . ∂ t θ ( t ) = � 2 W θ ( t ) Here J = i θ 1 � Z 1 � i θ 1 � Z 1 and A J , θ = A 11 θ 1 � Z 1 + A 11 θ 1 � Z 1 . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
The CR Yamabe Flow In particular, we start from the initial data with vanishing torsion : � ∂ t J ( t ) = 0 . ∂ t θ ( t ) = � 2 W θ ( t ) The CR Yamabe Flow (Chang-Chiu-Wu, 2010, Chang-Kuo, 2011) ∂ t θ ( t ) = � 2 W θ ( t ) . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Poincare Conjecture and Thurston Geometrization Conjecture via Ricci Flow Sphere and Torus decomposition Singularity formation Li-Yau gradient estimate for heat equation ( 1986) 1 Hamilton-Ivy curvature pinching estimate (1982, 1995) 2 Hamilton Harnack inequality ( 1982, 1988, 1993, etc) 3 Perelman entropy formulae and reduce distance (2002, 2003) 4 Geometric surgery by Hamilton and Perelman The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Geometrization problem of contact 3-manifolds Contact Decomposition theorem and Classi…cation CR Geometric and Analytic aspects : Existence of a " best possible geometric CR structure" on closed 1 contact 3-manifolds- spherical CR structure with vanishing torsion. 2 R ij : Ricci curvature tensor $ A 11 : pseudohermitian torsion Problem Sub-Laplacian ∆ b is degenerated along the missing dirction T by comparing the Riemannian Laplacian ∆ . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
geometrization problem of contact 3-manifolds Problem We proposed to deform any fixed CR structure under the torsion on a contact three dimensional space which shall break up due to the contact topological decomposition. Problem The asymptoic state of the torsion flow is expected to be broken up into pieces which satisfy the spherical CR structure with vanishing torsion. Problem The deformation will encounter singularities. The major question is to find a way to describe all possible singularities . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Pseudohermitian 3-manifold Let ( M , J , θ ) be the pseudohermitian 3-manifold. ( M , θ ) is a contact 3-manifold with θ ^ d θ 6 = 0. ξ = ker θ is called the 1 contact structure on M . A CR -structure compatible with ξ is a smooth endomorphism 2 J : ξ ! ξ such that J 2 = � identity . The CR structure J can extend to C � ξ and decomposes C � ξ into 3 the direct sum of T 1 , 0 and T 0 , 1 which are eigenspaces of J with respect to i and � i , respectively. The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Pseudohermitian 3-manifold Given a pseudohermitian structure ( J , θ ) : The Levi form h , i L θ is the Hermitian form on T 1 , 0 de…ned by 1 � � h Z , W i L θ = � i d θ , Z ^ W . The characteristic vector …eld of θ is the unique vector …eld T such 2 that θ ( T ) = 1 and L T θ = 0 or d θ ( T , � ) = 0. 1 g is the frame …eld for TM and f θ , θ 1 , θ ¯ 1 g is the Then f T , Z 1 , Z ¯ 3 coframe . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Pseudohermitian 3-manifold The pseudohermitian connection of ( J , θ ) is the connection r ψ . h . on TM � C (and extended to tensors) given by ¯ r ψ . h . Z 1 = ω 11 � Z 1 , r ψ . h . Z ¯ 1 , r ψ . h . T = 0 1 � Z ¯ 1 = ω ¯ 1 with d θ 1 = θ 1 ^ ω 11 + A 1 ¯ 1 θ ^ θ ¯ 1 ω 11 + ω ¯ 1 = 0 . ¯ 1 Di¤erentiating ω 11 gives d ω 11 = W θ 1 ^ θ ¯ 1 (mod θ ) where W is the Tanaka-Webster curvature. The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Pseudohermitian 3-manifold We can de…ne the covariant differentiations with respect to the pseudohermitian connection. 1 1 Z 1 f � ω 11 ( Z ¯ f , 1 = Z 1 f ; f 1 ¯ 1 = Z ¯ 1 ) Z 1 f . We de…ne the subgradient operator r b and the sublaplacian 2 operator ∆ b r b f = f , ¯ 1 Z 1 + f , 1 Z ¯ 1 , and ∆ b f = f , 1 ¯ 1 + f , ¯ 11 . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Pseudohermitian 3-manifold Example D is the strictly pseudoconvex domain D � C 2 and M = ∂ D with D = f r < 0 g and M = f r = 0 g . Choose ξ = TM \ J C 2 TM and θ = � i ∂ r j M with J = J C 2 j ξ . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Li-Yau Harnack Estimate Theorem (Li-Yau, 1986) The Li-Yau Harnack estimate ∂ ( ln u ) � jr ln u j 2 + m 2 t � 0 ∂ t for the positive solution u ( x , t ) of the time-independent heat equation ∂ u ( x , t ) = ∆ u ( x , t ) ∂ t in a complete Riemannian m-manifold with nonnegative Ricci curvature. The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Li-Yau-Hamilton Inequality Theorem ( Hamilton, 1993) Hamilton Harnack estimate (trace version) ∂ R ∂ t + R t + 2 r R � V + 2 Ric ( V , V ) � 0 for the Ricci ‡ow ∂ g ij ∂ t = � 2 R ij on Riemannian manifolds with positive curvature operator. The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Subelliptic Li-Yau gradient estimate Consider the heat equation ( L � ∂ ∂ t ) u ( x , t ) = 0 in a closed m -manifold with a positive measure and an operator with respect to the sum of squares of vector …elds l X 2 ∑ L = i , l � m , i = 1 where X 1 , X 2 , ..., X l are smooth vector …elds which satisfy Hörmander’s condition : the vector …elds together with their commutators up to …nite order span the tangent space at every point of M . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
Subelliptic Li-Yau gradient estimate Theorem (Cao-Yau, 1994) Suppose that [ X i , [ X j , X k ]] can be expressed as linear combinations of X 1 , X 2 , ..., X l and their brackets [ X 1 , X 2 ] , ..., [ X l � 1 , X l ] . Then, for the positive solution u ( x , t ) of heat ‡ow on M � [ 0 , ∞ ) , there 000 and 0 , C 00 , C 2 < λ < 2 1 3 , such that for any δ > 1 , exist constants C f ( x , t ) = ln u ( x , t ) satis…es the following gradient estimate 0 ( 1 + j Y α f j 2 ) λ � C 00 + C j X i f j 2 � δ f t + ∑ 000 t λ ∑ t + C λ � 1 α i with f Y α g = f [ X i , X j ] g . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
CR Li-Yau gradient estimate By choosing a frame f T , Z 1 , Z ¯ 1 g of TM � C with respect to the Levi form and f X 1 , X 2 g such that J ( Z 1 ) = iZ 1 and J ( Z 1 ) = � iZ 1 and Z 1 = 1 2 ( X 1 � iX 2 ) and Z 1 = 1 2 ( X 1 + iX 2 ) , it follows that and ∆ b = 1 2 ) = 1 2 ( X 2 1 + X 2 [ X 1 , X 2 ] = � 2 T 2 L . Note that W ( Z , Z ) = Wx 1 x ¯ 1 x ¯ 1 x ¯ 1 1 ) and Tor ( Z , Z ) = 2 Re ( iA ¯ 1 ¯ for all Z = x 1 Z 1 2 T 1 , 0 . The 10th Paci…c Rim Geometry Conference O Shu-Cheng Chang (National Taiwan University) CR Li-Yau Gradient Estimate and Perelman Entropy Formulae / 52
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