Some New Myers-Type Theorems via m-Bakry-mery Ricci Curvature - PowerPoint PPT Presentation

Some New Myers-Type Theorems via m-Bakry-Émery Ricci Curvature Homare TADANO Tokyo University of Science, JAPAN tadano@rs.tus.ac.jp Symmetry and Shape Celebrating the 60th birthday of Prof. J. Berndt October 30, 2019 University of Santiago de Compostela, SPAIN

Aim & Plan 1. Introduction : A Brief Review of — The Classical Bonnet-Myers Theorem 2. Previous Works and New Results : — m-Modified Ricci and m-Bakry-Émery Ricci Curvatures — Bonnet-Myers Type Theorems — New Compactness Theorems

Bonnet-Myers Theorem Natural questions about a complete Riemannian manifold are (1) When is compact ? ( M, g ) (2) How large is ? diam( M, g ) Theorem (S. B. Myers 1942) If s.t. ∃ λ > 0 Ric g > λ g r n − 1 ( M, g ) : compact & . diam( M, g ) 6 π ⇒ = λ A topological obstruction to the existence of a metric with a positive Ricci curvature bound.

Modified Ricci and Bakry-Émery Ricci Curvatures : vector field, : smooth function. V ∈ X ( M ) f ∈ C ∞ ( M ) Definition (D. Bakry and M. Émery 1985, M. Limoncu 2009) Ric V := Ric g + 1 : modified Ricci curvature 2 L V g : Bakry-Émery Ricci curvature Ric f := Ric g + Hess f Good substitutes of the Ricci curvature : eigenvalue estimates, Li-Yau Harnack inequalities, … Question Is the Myers theorem true for Ric V and Ric f ? Remark The shrinking Gaussian soliton is non-compact.

A Bonnet-Myers Type Theorem via 
 Modified Ricci Curvature Theorem (M. Fernández-Lópes and E. García-Río 2004, M. Limoncu 2009, — 2015, J.-Y. Wu 2017) Suppose s.t. ∃ λ > 0 Ric V := Ric g + 1 . 2 L V g > λ g If for | V | 6 k ∃ k > 0 r diam( M, g ) 6 2 k n − 1 ( M, g ) : compact & . λ + π ⇒ = λ Remark , may be improved to | V | 6 k | V | ( x ) 6 ∃ α r ( x ) + ∃ β where 0 6 α < λ and β ∈ R .

A Bonnet-Myers Type Theorem via Bakry-Émery Ricci Curvature Remark A function does not always satisfy f ∈ C ∞ ( M ) . | r f | 6 9 k ( ) | f | 6 9 k Theorem (G. Wei and W. Wylie 2007, M. Limoncu 2009, — 2015) Suppose s.t. ∃ λ > 0 . Ric f := Ric g + Hess f > λ g If for | f | 6 k ∃ k > 0 r 8 k π ( M, g ) : compact & . diam( M, g ) 6 π + n − 1 ⇒ = √ λ

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> A New Compactness Theorem via Bakry-Émery Ricci Curvature Theorem ( — 2018) for ∃ k > 0 . Suppose | f | 6 k If ∃ p ∈ M , ∃ r 0 > 0 , s.t. ∃ ` > 2 C ( r 0 , ` ) Ric f ( x ) > ( n + 4 k − 1) ( r 0 + r ( x )) ` g ( x ) for ∀ x ∈ M , where ( ` − 1) ` ( ( ` − 2) ` − 2 r ` − 2 ` > 2 , 0 C ( r 0 , ` ) := 1 + " , ∀ " > 0 ` = 2 ( M, g ) : compact. ⇒ = Remark This theorem was proved by J. Wan (2017) via . Ric g

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> m-Modified Ricci and m-Bakry-Émery Ricci Curvatures : vector field, : smooth function, V ∈ X ( M ) f ∈ C ∞ ( M ) . m ∈ R ∪ {± ∞ } Definition (D. Bakry and M. Émery 1985, M. Limoncu 2009) V := Ric g + 1 1 m � nV ∗ ⌦ V ∗ Ric m Ric n ( m 6 = n ) , V := Ric g 2 L V g � : m-modified Ricci curvature 1 Ric m Ric n f := Ric g + Hess f � ( m 6 = n ) , f := Ric g f ⌦ d m � nd f : m-Bakry-Émery Ricci curvature (1) Good substitutes of the Ricci curvature. (2) Important in Optimal Transport Theory by Lott-Sturm-Villani and in Perelman’s entropy formula for the Ricci flow.

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> A Bonnet-Myers Type Theorem via m-Modified Ricci and m-Bakry-Émery Ricci Curvatures Theorem (Z. Qian 1995, M. Limoncu 2009, K. Kuwada 2011) Let If . s.t. m > n ∃ λ > 0 V := Ric g + 1 1 m − nV ∗ ⊗ V ∗ > λ g . Ric m 2 L V g − r m − 1 ( M, g ) : compact & . diam( M, g ) 6 π ⇒ = λ A topological obstruction to the existence of a metric with a positive m-modified Ricci curvature bound.

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> A New Compactness Theorem via m-Modified Ricci and m-Bakry-Émery Ricci Curvatures Theorem ( — 2018) . Suppose ∃ p ∈ M , ∃ r 0 > 0 , Let s.t. m > n ∃ ` > 2 C ( r 0 , ` ) Ric m V ( x ) > ( m − 1) ( r 0 + r ( x )) ` g ( x ) for ∀ x ∈ M , where ( ` − 1) ` ( ( ` − 2) ` − 2 r ` − 2 ` > 2 , 0 C ( r 0 , ` ) := 1 + " , ∀ " > 0 ` = 2 ( M, g ) : compact. ⇒ = Remark This theorem was proved by J. Wan (2017) via . Ric g