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

Symmetry and Shape Celebrating the 60th birthday of Prof. J. Berndt October 30, 2019 University of Santiago de Compostela, SPAIN

Some New Myers-Type Theorems via m-Bakry-Émery Ricci Curvature

Homare TADANO Tokyo University of Science, JAPAN tadano@rs.tus.ac.jp

Aim & Plan

• 1. Introduction : A Brief Review of
• 2. Previous Works and New Results :

— m-Modified Ricci and m-Bakry-Émery Ricci Curvatures — Bonnet-Myers Type Theorems — The Classical Bonnet-Myers Theorem — New Compactness Theorems

Bonnet-Myers Theorem

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

Modified Ricci and Bakry-Émery Ricci Curvatures

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

Theorem (M. Fernández-Lópes and E. García-Río 2004, Suppose s.t. ∃ λ > 0

• M. Limoncu 2009, — 2015, J.-Y. Wu 2017)

RicV := Ricg + 1 2LV g > λg If (M, g) : compact & = ⇒ , for Remark may be improved to . diam(M, g) 6 2k λ + π r n − 1 λ |V | 6 k |V | 6 k ∃ k > 0 . where 0 6 α < λ and β ∈ R. |V |(x) 6 ∃ αr(x) + ∃ β

A Bonnet-Myers Type Theorem via 
 Modified Ricci Curvature

Theorem (G. Wei and W. Wylie 2007, M. Limoncu 2009, — 2015) s.t. ∃ λ > 0 . If for Ricf := Ricg + Hessf > λg = ⇒ Remark |rf| 6 9 k ( ) |f| 6 9 k f ∈ C∞(M) A function does not always satisfy . ∃ k > 0 |f| 6 k Suppose . diam(M, g) 6 π √ λ r 8k π + n − 1 (M, g) : compact &

A Bonnet-Myers Type Theorem via Bakry-Émery Ricci Curvature

Theorem ( — 2018) If ∃ p ∈ M , ∃ r0 > 0 , s.t. , where for ∀ x ∈ M (M, g) : compact. = ⇒

A New Compactness Theorem via Bakry-Émery Ricci Curvature

for ∃ k > 0 |f| 6 k Suppose ∃ ` > 2

Ricf(x) > (n + 4k − 1) C(r0, `) (r0 + r(x))` g(x)

C(r0, `) := (

(`−1)` (`−2)`−2 r`−2

` > 2, 1 + ", ∀" > 0 ` = 2

. Remark This theorem was proved by J. Wan (2017) via . Ricg

m-Modified Ricci and m-Bakry-Émery Ricci Curvatures

V ∈ X(M) f ∈ C∞(M) : vector field, : smooth function, Definition (D. Bakry and M. Émery 1985, M. Limoncu 2009) : m-modified Ricci curvature : m-Bakry-Émery Ricci curvature . (2) Important in Optimal Transport Theory by Lott-Sturm-Villani and in Perelman’s entropy formula for the Ricci flow. (1) Good substitutes of the Ricci curvature. m ∈ R ∪ {±∞}

Ricm

V := Ricg + 1

2LV g 1 m nV ∗ ⌦ V ∗ (m 6= n), Ricn

V := Ricg

Ricm

f := Ricg + Hessf

1 m nd f ⌦ d f (m 6= n), Ricn

f := Ricg

Theorem (Z. Qian 1995, M. Limoncu 2009, K. Kuwada 2011) A topological obstruction to the existence of

A Bonnet-Myers Type Theorem via m-Modified Ricci and m-Bakry-Émery Ricci Curvatures

a metric with a positive m-modified Ricci curvature bound. If s.t. ∃ λ > 0 Let . Ricm

V := Ricg + 1

2LV g − 1 m − nV ∗ ⊗ V ∗ > λg (M, g) : compact & = ⇒ . diam(M, g) 6 π r m − 1 λ . m > n

A New Compactness Theorem via m-Modified Ricci and m-Bakry-Émery Ricci Curvatures

Theorem ( — 2018) Let . Suppose ∃ p ∈ M , ∃ r0 > 0 , s.t. , where for ∀ x ∈ M (M, g) : compact. = ⇒ Remark This theorem was proved by J. Wan (2017) via . Ricg m > n

C(r0, `) := (

(`−1)` (`−2)`−2 r`−2

` > 2, 1 + ", ∀" > 0 ` = 2

Ricm

V (x) > (m − 1)

C(r0, `) (r0 + r(x))` g(x)

∃ ` > 2