Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Let $P^g$ be a positive, self-adjoint elliptic differential operator on $(M,g)$. Then we have it that $(u,P^g v)=({P^g}^{1/2}u,{P^g}^{1/2}v)=\int_M {P^g}^{1/2}u {P^g}^{1/2}v dv_g$ is an inner-product on $H^2(M)$. Now let $g^n$ be a sequence of Riemannian metrics on $M$ that converges to $g$ in $C^4$ norm. Let $u^n$ and $v^n$ be a sequence of smooth functions that converge in $H^2$ norm to $u$ and $v$, respectively. Suppose $(u^n,P^{g^n} v^n) \rightarrow 0$, with $(u^n,P^{g^n} v^n) \neq 0$, as $n \rightarrow \infty$. Suppose as well that $(u^n,P^g v^n) \rightarrow 0$ as $n \rightarrow \infty$. Does it follow that $(u^n,P^g v^n)$ over (u^n,P^{g^n} $(u^n,P^{g^n} v^n)$ goes to 1 as $n \rightarrow \infty$.infty$?
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Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Let $P_g$ P^g$ be a positive, self-adjoint elliptic differential operator on $(M,g)$. Then we have it that $==\int_M P_g^{1/2}u P_g^{1/2}v (u,P^g v)=({P^g}^{1/2}u,{P^g}^{1/2}v)=\int_M {P^g}^{1/2}u {P^g}^{1/2}v dv_g$ is an inner-product on $H^2(M)$. Now let $g_n$ g^n$ be a sequence of Riemannian metrics on $M$ that converges to $g$ in $C^4$ norm. Let $u_n$ u^n$ and $v_n$ v^n$ be a sequence of smooth functions that converge in $H^2$ norm to $u$ and $v$, respectively. Suppose $(u^n,P^{g^n} v^n) \rightarrow 0$, with $(u^n,P^{g^n} v^n) \neq 0$, as $n \rightarrow \infty$. Suppose as well that $(u^n,P^g v^n) \rightarrow 0$ as $n \rightarrow \infty$. Does it follow that $$(u^n,P^g v^n)$ over $ (u^n,P^{g^n} v^n)$ goes to 1 as $n \rightarrow \infty$. |
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Hilbert space inner products with converging metricsLet $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Let $P_g$ be a positive, self-adjoint elliptic differential operator on $(M,g)$. Then we have it that $==\int_M P_g^{1/2}u P_g^{1/2}v dv_g$ is an inner-product on $H^2(M)$. Now let $g_n$ be a sequence of Riemannian metrics on $M$ that converges to $g$ in $C^4$ norm. Let $u_n$ and $v_n$ be a sequence of smooth functions that converge in $H^2$ norm to $u$ and $v$, respectively. Suppose $ \rightarrow 0$, with $ \neq 0$, as $n rightarrow \infty$. Suppose as well that $ \rightarrow 0$ as $n \rightarrow \infty$. Does it follow that $$ over $ goes to 1 as $n \rightarrow \infty$.
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