General Reilly-type inequalities for submanifolds of weighted Euclidean spaces (Q2804281)

Let \((M^n,g)\) be a connected and oriented closed Riemannian manifold isometrically immersed into the Euclidean space \(\mathbb{R}^N\) endowed with a density \(e^{-f}\). For a positive symmetric divergence-free \((1,1)\)-tensor \(T\), the operator \(L_{T,f}\) is defined as NEWLINE\[NEWLINEL_{T,f}u=-\operatorname{div}_f(T\nabla u)=-\operatorname{div}(T\nabla u)+\langle \nabla f,T\nabla u\rangleNEWLINE\]NEWLINE NEWLINEfor any function \(u\in C^2(M)\). When \(T=\mathrm{Id}\), \(L_{T,f}\) is just the Bakry-Émery Laplacian. In this paper, the author derived the new upper bound for the first positive eigenvalue \(\lambda_1\) of the operator \(L_{T,f}\). Precisely, if \(S\) is also a symmetric divergence-free \((1,1)\)-tensor over \(M\), then NEWLINENEWLINE\[NEWLINE \lambda_1\left(\int_M\mathrm{tr}(S)\mu_f\right)^2\leq \left(\int_M\mathrm{tr}(T)\mu_f\right)\int_M\left(\|H_S\|^2+\|S\nabla f\|^2\right)\mu_f, \eqno{(1)}NEWLINE\]NEWLINE NEWLINEwhere \(H_S\) is defined as NEWLINE\[NEWLINEH_S=\sum_{i,j=1}^nS(e_i,e_j)B(e_i,e_j)NEWLINE\]NEWLINE for a local orthonormal frame of \(TM\) and \(B\) is the second fundamental form of \(M\). The proof of (1) is done by applying the coordinate functions as the test functions in the Rayleigh quotient and employing the newly proved weighted Hsiung-Minkowski formula. In the special case \(S=\mathrm{Id}\), the equality case of (1) is also investigated in this paper.