본문 바로가기
KERD
논문KCI 등재

Rigidity characterization of compact Ricci solitons

Jian Zhou|Fengjiang Li
대한수학회 (The Korean Mathematical Society)|2019.11|대한수학회지 (Journal of the KMS)|제56권 제6호|pp. 1475-1488|DOI 10.4134/JKMS.j180747

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

Ricci solitonEinstein manifoldRicci mean valueWeyl conformal curvatue tensor
주제분류수학

초록

In this paper, we firstly define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton $(M^{n},g) (n \geq 3)$ is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

Abstract

In this paper, we firstly define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton $(M^{n},g) (n \geq 3)$ is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

참고문헌 (24)

  1. Christoph Böhm, "Manifolds with positive curvature operators are space forms", Annals of Mathematics, 167(3), pp.1079~1097 (2008), Annals of Mathematics[DOI]
  2. H.-D. Cao, "Existence of gradient Kahler-Ricci solitons", Elliptic and parabolic methods in geometry, pp.1~16 (1994), A K Peters
  3. Huai-Dong Cao, "On locally conformally flat gradient steady Ricci solitons", Transactions of the American Mathematical Society, 364(5), pp.2377~2391 (2012), American Mathematical Society (AMS)[DOI]
  4. Xiaodong Cao, "Compact gradient shrinking Ricci solitons with positive curvature operator", Journal of Geometric Analysis, 17(3), pp.425~433 (2007), Springer Science and Business Media LLC[DOI]
  5. XIAODONG CAO, "ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS", Communications in Contemporary Mathematics, 13(02), pp.269~282 (2011), World Scientific Pub Co Pte Lt[DOI]
  6. Xiuxiong Chen, "A note on uniformization of Riemann surfaces by Ricci ow", Proceedings of the American Mathematical Society, 134(11), pp.3391~3394 (2006), American Mathematical Society (AMS)[DOI]
  7. Shiu-Yuen Cheng, "Hypersurfaces with constant scalar curvature", Mathematische Annalen, 225(3), pp.195~204 (1977), Springer Science and Business Media LLC[DOI]
  8. B. Chow, "The Ricci Flow: Techniques and Applications. Part I" (2007), American Mathematical Society
  9. B. Chow, "The Ricci Flow: An Introduction" (2004), American Mathematical Society
  10. A. Derdzinski, "Compact Ricci solitons", Work in progress, 1~57 (2009)
  11. Manolo Eminenti, "Ricci solitons: the equation point of view", manuscripta mathematica, 127(3), pp.345~367 (2008), Springer Science and Business Media LLC[DOI]
  12. Z. Guo, "Scalar curvature of self-shrinker", J. Math. Soc. Japan, 70(3), pp.1103~1110 (2018)[DOI]
  13. D. Gilbarg, "Elliptic Partial Differential Equations of Second Order" (1977), Springer-Verlag
  14. R. S. Hamilton, "Three-manifolds with positive Ricci curvature", J. Differential Geom, 17(2), pp.255~306 (1982)[DOI]
  15. R. S. Hamilton, "The Ricci ow on surfaces", Mathematics and general relativity, Contemp. Math, 71, pp.237~262 (1988), Amer. Math. Soc
  16. R. S. Hamilton, "The Harnack estimate for the Ricci ow", J. Differential Geom, 37(1), pp.225~243 (1993)[DOI]
  17. E. Hopf, "Elementare Bemerkungen uber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus", pp.147~152 (1927), Sitzungsberichte Akad
  18. Thomas Ivey, "Ricci solitons on compact three-manifolds", Differential Geometry and its Applications, 3(4), pp.301~307 (1993), Elsevier BV[DOI]
  19. N. Koiso, "Recent topics in differential and analytic geometry", pp.327~337 (1990), Academic Press
  20. Aaron Naber, "Noncompact shrinking four solitons with nonnegative curvature", Journal für die reine und angewandte Mathematik (Crelles Journal), 2010(645), pp.125~153 (2010), Walter de Gruyter GmbH[DOI]
  21. Lei Ni, "On a classification of gradient shrinking solitons", Mathematical Research Letters, 15(5), pp.941~955 (2008), International Press of Boston[DOI]
  22. Masafumi Okumura, "Hypersurfaces and a Pinching Problem on the Second Fundamental Tensor", American Journal of Mathematics, 96(1), pp.207~213 (1974), JSTOR[DOI]
  23. G. Perelman, "The entropy formula for the Ricci ow and its geometric applications", Mathematics, 95, pp.1~39 (2002)
  24. Zhu-Hong Zhang, "Gradient shrinking solitons with vanishing Weyl tensor", Pacific Journal of Mathematics, 242(1), pp.189~200 (2009), Mathematical Sciences Publishers[DOI]

관련 논문AI추천

Gradient almost Ricci solitons with vanishing conditions on Weyl tensor and Bach tensor고진석, 황승수대한수학회 · 대한수학회지2020 · 제57권 제2호Rigidity of gradient shrinking and expanding Ricci solitonsFei Yang, Liangdi Zhang대한수학회 · 대한수학회보2017 · 제54권 제3호The rigidity of almost Ricci-Yamabe solitons with concurrent-recurrent vector fields대한수학회 · 대한수학회보2026 · 제63권 제1호Gradient Ricci solitons with half harmonic Weyl curvature and two Ricci eigenvalues강유태, 김종수대한수학회 · 대한수학회논문집2022 · 제37권 제2호Some results in $\eta$-Ricci soliton and gradient $\rho$-Einstein soliton in a complete Riemannian manifoldAbsos Ali Shaikh, Chandan Kumar Mondal대한수학회 · 대한수학회논문집2019 · 제34권 제4호Some Triviality Characterizations on Gradient Almost Yamabe SolitonsMampi Howlader, Puja Sarkar, Uday Chand De경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2023 · 제63권 제4호

이 간행물의 다른 논문 (12건)

Evolution and monotonicity for a class of quantities along the Ricci-Bourguignon flowFarzad Daneshvar, Asadollah Razavipp. 1441-1461Number of weak Galois-Weierstrass points with Weierstrass semigroups generated by two elementsTakeshi Takahashi, Jiryo Komedapp. 1463-1474Weak normal properties of partial isometriesYanying Men, Sen Zhu, Ting Liupp. 1489-1502Strong preservers of symmetric arctic rank of nonnegative real matricesLuis Hernandez Encinas, 송석준, 비어슬리pp. 1503-1514Embedding distance graphs in finite field vector spacesAlex Iosevich, Hans Parshallpp. 1515-1528
원문 보기