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In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in \cite{PhDJingMao}. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the $p$-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.
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참고문헌 (21)
참고문헌 신청
B. Andrews / 1994 / Contraction of convex hypersurfaces in Euclidean space / Calc. Var. Partial Differential Equations 2(2):151~171
E. Cabezas-Rivas / 2010 / Volume-preserving ow by powers of the mth mean curvature / Calc. Var. Partial Differential Equations 38(3-4):441~469
X. Cao / 2007 / Eigenvalues of (-Δ + R/2 ) on manifolds with nonnegative curvature operator / Math. Ann 337(2):435~441
X. Cao / 2008 / First eigenvalues of geometric operators under the Ricci ow / Proc. Amer. Math. Soc 136(11):4075~4078
X. Cao / 2012 / Estimate and monotonicity of the first eigenvalue under the Ricci flow / Math. Ann 354(2):451~463
E. Cabezas-Rivas / 2010 / Volume-preserving ow by powers of the mth mean curvature / Calc. Var. Partial Differential Equations 38(3-4):441~469
X. Cao / 2007 / Eigenvalues of (-Δ + R/2 ) on manifolds with nonnegative curvature operator / Math. Ann 337(2):435~441
X. Cao / 2008 / First eigenvalues of geometric operators under the Ricci ow / Proc. Amer. Math. Soc 136(11):4075~4078
X. Cao / 2012 / Estimate and monotonicity of the first eigenvalue under the Ricci flow / Math. Ann 354(2):451~463
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