The authors revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds.By building on classical resultslike Li-Yau's and Yang's inequalities,they derive upper...The authors revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds.By building on classical resultslike Li-Yau's and Yang's inequalities,they derive upper and lower bounds for eigenvalues.For the projective spaces and their minimal submanifolds,they also give explicit estimates on the lower bound for the eigenvalue of the Dirichlet Laplacian.展开更多
In this paper,we first construct compact embeddedλ-hypersurfaces with the topology of torus which are calledλ-torus in Euclidean spacesℝn^+1.Then,we give many compact immersedλ-hypersurfaces in Euclidean spacesℝn^+1.
In this paper,we completely classify 3-dimensional complete self-shrinkers with the constant norm S of the second fundamental form and the constant f3in the Euclidean space R^(4),where hij’s are components of the sec...In this paper,we completely classify 3-dimensional complete self-shrinkers with the constant norm S of the second fundamental form and the constant f3in the Euclidean space R^(4),where hij’s are components of the second fundamental form,S=∑i,jhij2and f3=∑i,j,khijhjkhki.展开更多
基金supported by the National Natural Science Foundation of China(No.11831005,1257010742)andthe NSFC-FWO grant(No.11961131001)the JSPS Grant-in-Aidfor Scientific Research(No.25K06992)+1 种基金the MEXT Promotion of Distinctive Joint Research Center Program(No.JPMXP0723833165)the Osaka Metropolitan University Strategic Research Promotion Project(Development of International Research Hubs).
文摘The authors revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds.By building on classical resultslike Li-Yau's and Yang's inequalities,they derive upper and lower bounds for eigenvalues.For the projective spaces and their minimal submanifolds,they also give explicit estimates on the lower bound for the eigenvalue of the Dirichlet Laplacian.
基金supported by JSPS Grant-in-Aid for Scientific Research(B)(Grant No.16H03937)Challenging Exploratory Research+1 种基金supported by National Natural Science Foundation of China(Grant No.11771154)by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2018)。
文摘In this paper,we first construct compact embeddedλ-hypersurfaces with the topology of torus which are calledλ-torus in Euclidean spacesℝn^+1.Then,we give many compact immersedλ-hypersurfaces in Euclidean spacesℝn^+1.
基金supported by Japan Society for the Promotion of Science(JSPS)Grant-in-Aid for Scientific Research(Grant Nos.16H03937 and 22K03303)the Fund of Fukuoka University(Grant No.225001)+3 种基金supported by China Postdoctoral Science Foundation(Grant No.2022M711074)supported by National Natural Science Foundation of China(Grant No.12171164)Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2018)Guangdong Natural Science Foundation(Grant No.2023A1515010510)。
文摘In this paper,we completely classify 3-dimensional complete self-shrinkers with the constant norm S of the second fundamental form and the constant f3in the Euclidean space R^(4),where hij’s are components of the second fundamental form,S=∑i,jhij2and f3=∑i,j,khijhjkhki.
