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Asymptotics of spectral gaps on loop spaces

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Shigeki Aida
University of Tokyo
Mon, 2017-09-04 16:30 - 17:30
MPIM Lecture Hall

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Let $x_0, y_0\in M$ and consider the loop space $P_{x_0,y_0}(M)=\{\gamma\in C([0,1]\to  M)~|~\gamma(0)=x_0, \gamma(1)=y_0\}$. Let $\nu^{\lambda}$ be the pinned measure defined by the transition probability $p(t/\lambda,x,y)$, where $p(t,x,y)$ denotes the heat kernel of the diffusion semigroup $e^{t\Delta/2}$.  Heuristically, we have
$$ d\nu^{\lambda}_{x_0,y_0}(\gamma)=\frac{1}{Z_{\lambda}} \exp\left(-\lambda E(\gamma)\right) d\gamma, $$
where $\mathcal{E}(\gamma)=\frac{1}{2}\int_0^1\left|\gamma'(t)\right|^2dt$ and $d\gamma$ is the ``infinite dimensional Riemannian measure''. Let $D$ be the $H$-derivative on $P_{x_0,y_0}$ and consider the Dirichlet form,
$$ \mathcal{E}^{\lambda}(F,F)=\int_{P_{x_0,y_0}(M)} |(DF)(\gamma)|_{\mathrm{H_o}}^2d\nu^{\lambda}_{x_0,y_0}(\gamma). $$
Assume that $M$ is a Riemannian manifold with a pole $y_0$. Then, the function $E$ defined on the $H^1$ subset of $P_{x_0,y_0}(M)$ is a Morse function and the unique critical point is the minimal geodesic between $x_0$ and $y_0$. Under additional assumptions, we determine the asymptotic behavior of the spectral gap of $\mathcal{E}^{\lambda}$.

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