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In this paper, we provide comparisons of solutions to nonlinear parabolic equations on complete manifolds with R i c ≥ (n - 1) κ , where κ = 0 or 1, which extend the results of Cheng et al. (Schwarz symmetrizations in parabolic equations on complete manifolds, 2021) and Vázquez (C R Acad Sci Paris Ser I Math 295(2):71–74, 1982). As an application, we give an alternate proof to Faber–Krahn inequality for Dirichlet Laplacian on these manifolds, which was proved in Balogh and Kristály (Math Ann , 2022) [see also Chen and Li (J Geom Anal 33(4):123, 2023) and Fogagnolo and Mazzieri (J Funct Anal 283:109638, 2022)]. Additionally, we also obtain corresponding comparison results on minimal submanifolds in Riemannian manifolds with nonnegative sectional curvature. [ABSTRACT FROM AUTHOR]

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber–Krahn inequality for the first eigenvalue of Dirichlet Laplacian, L 1 - and L ∞ -moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian. [ABSTRACT FROM AUTHOR]

We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi–Montiel–Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any C 3 , α conformal compactification of some Poincare–Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any C 3 , α conformal compactification of some Poincare–Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant. In two cases the Poincare–Einstein metrics are standard hyperbolic metric. [ABSTRACT FROM AUTHOR]

In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere $$S^{n+1}(1)$$ . We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, $$\varphi :\ M \rightarrow S^{n+1}(1)$$ is isometric to $$S^1(r)\times S^{n-1}(\sqrt{1-r^2})$$ when $$H\ne 0$$ or $$\varphi :\ M \rightarrow S^{n+1}(1)$$ is isometric to a Clifford torus $$ S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right) $$ , for $$k=1, 2, \ldots , n-1$$ when $$H=0$$ . [ABSTRACT FROM AUTHOR]

We study the Dirac-Witten operator for spacelike spin submanifolds in pseudo-Riemannian manifolds. When the normal bundles are spin and odd-dimensional, we derive new eigenvalue lower bounds. We also prove the generalized positive mass theorem using this operator. [ABSTRACT FROM AUTHOR]

We consider the question of characterizing compact quotients of the complex 2-ball by curvature conditions, which improve the known results. Moreover, we also give curvature conditions such that a compact Kähler-Einstein surface is bi-holomorphic to a locally symmetric space. [ABSTRACT FROM AUTHOR]

For n ≥ 7, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact n-dimensional submanifolds in an ( n + p)-dimensional space form in terms of the mean curvature and the Q-curvature. [ABSTRACT FROM AUTHOR]

Given a compact Riemannian manifold M, we consider a warped product manifold M¯=I×hM, where I is an open interval in R. For a positive function ψ defined on M¯, we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287-1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535), to obtain the curvature estimates for Hessian equations σk(κ)=ψ(V,ν(V)). We also obtain some existence results for the starshaped compact hypersurface Σ satisfying the above equation with various assumptions. [ABSTRACT FROM AUTHOR]