# Principal eigenvalue of Laplacian under volume preserving mean curvature flow

Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some diffeomorphism $$F_0: \mathbb R^n \supset U \rightarrow F_0(U) \subset M_0 \subset \mathbb R^{n+1}$$ We deform it by volume preserving mean curvature flow \begin{align} &\partial_t F(x,t) = (h(t)-H(x,t))\cdot \nu(x,t) ~~~~~x\in U , t\ge 0 \\ &F(\cdot, 0) =F_0 \end{align} where $\nu$ is out normal vector , $H$ is mean curvature , and $h(t)$ is the average of the mean curvature on $M_t$ $$h(t)=\frac{\int_{M_t} H d\mu}{\int_{M_t} d\mu}$$ As Huisken, Gerhard, The volume preserving mean curvature flow, J. Reine Angew. Math. 382, 35-48 (1987). ZBL0621.53007. We know the $M_t$ converge to a round sphere enclosing the same volume as $M_0$.

Let $V(t)$ be the domain enclosed by $M_t$. Consider the eigenvalue question. \begin{align} &-\Delta f=\lambda f &x\in V(t) \\ &f=0 &x\in M_t \end{align} Let $\lambda$ be the first positive eigenvalue. By the Huisken's theory, we know $\lambda(V(t))$ converge to $\lambda(B)$, $B$ is the ball has same volume with $V(0)$. And as in this question , we know $\lambda(V(t))\ge\lambda(B)$. So, maybe, we have $$\frac{d\lambda(V(t))}{dt} \le 0$$ when $t$ is large enough. But I don't know how to show it. In fact , I do some calculation , and get $$\frac{d\lambda}{dt}=\int_{M_t} \partial_t f \nabla f\cdot \nu dS$$ $f$ is the eigenfunction of $\lambda$ and $\int_{V(t)} f^2 =1$. $\nu$ is the out normal vector. But,I don't know how to esitimate $\partial_t f$ and $\nabla f$.

• Can it be obtained by the monotonicity of the isoperimetric ratio and something like Faber Krahn inequality? – Paul Bryan Apr 5 '17 at 2:01
• encyclopediaofmath.org/index.php/… – lanse7pty Apr 13 '17 at 12:53
• @PaulBryan Could you talk about how to get the monotonicity of the isoperimetric ratio ? In my view , I guess that is from Faber Krahn inequality. But I don't know the $\frac{d\lambda(V(t))}{dt} \le 0$ for all $t$. – lanse7pty Apr 13 '17 at 12:56
• I was thinking that the isoperimetric ratio $A^{n+1}/V^n$ is monotone decreasing along the flow and perhaps via something lie Faber-Krahn the monotonicty could be transferred to $\lambda_1$. – Paul Bryan Apr 14 '17 at 8:51