Manifolds with many small wormholes: norm resolvent and spectral convergence

Abstract

We present results concerning the norm convergence of resolvents for wildperturbations of the Laplace-Beltrami operator. This article is a continuation of ouranalysis on wildly perturbed manifolds presented in [AP21]. We study here manifoldswith an increasing number of small (i.e., short and thin) handles added. The handlescan also be seen as wormholes, as they connect different parts being originally far away.We consider two situations: if the small handles are distributed too sparse the limitoperator is the unperturbed one on the initial manifold, the handles are fading. Onthe other hand, if the small handles are dense in certain regions the limit operator isthe Laplace-Beltrami operator acting on functions which are identical on the two partsjoined by the handles, the handles hence produce adhesion. Our results also apply tonon-compact manifolds. Our work is based on a norm convergence result for operatorsacting in varying Hilbert spaces described in the book [P12] by the second author.

Figures (12)

• Figure 1: The manifold $M_\varepsilon$ obtained from $X$ (here the top and lower flat region) by removing many small balls $B_\varepsilon(p)$ of radius $\varepsilon$ by attaching handles $C_\varepsilon(p)$. The small dotted lines mean identification.
• Figure 2: The range of the parameters $\alpha$ ($\eta_\varepsilon=\varepsilon^\alpha r_0$, the larger the denser the handles are) and $\lambda$ ($\ell_\varepsilon=\varepsilon^\lambda$, the larger the shorter the handles are) in which Corollary \ref{['cor:handles0']} is valid ($m \ge 3$). For the letters describing points in the $(\alpha,\lambda)$-plane, see \ref{['eq:alpha.lambda.points']}). The estimate on the convergence speed is of order $\mathrm O(\varepsilon^{(\alpha_m-\alpha)/(2(1-\alpha_m))})$ in the (infinite) rectangle $C'G(\alpha_m^*,\infty)(\alpha_m,\infty)$ (lightest grey), of order $\mathrm O(\varepsilon^{(\alpha_m+\alpha)/2)})$ in the (infinite) rectangle $D^-(0,\infty)(\alpha_m^*,\infty)G$ (middle grey) resp. of order $\mathrm O(\varepsilon^\lambda)$ in the triangle $C'DD^-G$ (dark grey).
• Figure 3: The range of the parameters $\alpha$ ($\eta_\varepsilon=\varepsilon^\alpha r_0$ in which Corollary \ref{['cor:handles1']} is valid. The estimate on the convergence speed is of order $\mathrm O(\varepsilon^{(1-\lambda)/2})$ in the polygon $C'DD^-G$ (horizontal lines); of order $\mathrm O(\varepsilon^{(-\lambda+(m-1)-m\alpha)/2})$ in the polygon (diagonal lines) resp. of order $\mathrm O(\varepsilon^\lambda)$ (dark grey).
• Figure 4: Comparison of the two fading results: horizontal light lines: only Corollary \ref{['cor:handles1']} applies; vertical light lines: only Corollary \ref{['cor:handles0']} applies; horizontal dark lines: Corollary \ref{['cor:handles1']} has better estimates than Corollary \ref{['cor:handles0']}; vertical dark lines: Corollary \ref{['cor:handles0']} has better estimates than Corollary \ref{['cor:handles1']}; grey area: both results have the same order on the convergence speed. Dotted area (only for $m\ge 5$): none of our results apply.
• Figure 5: Left: The manifold $M_\varepsilon$ with handles of radius $\varepsilon$ and length $\ell_\varepsilon$; medium grey: $\Omega^+$ (top) and $\Omega^-$ (bottom), together with its $\widetilde{\varepsilon}$-neighbourhood $\Omega^+_{\widetilde{\varepsilon}}$ and $\Omega^+_{\widetilde{\varepsilon}}$ in medium and dark grey, respectively. Right: The space after identification: one can think of the limit space as a piece of paper where the parts $\Omega^+$ (top) and $\Omega^-$ (bottom) are glued together. The dotted line in the right figure is used to illustrate the effect of the symmetrisation operator $S f$ in Figure \ref{['fig:symmetrisation']} for the proof of Theorem \ref{['thm:handles3']}.

Theorems & Definitions (59)

• Definition 2.1: non-concentrating property
• Proposition 2.2: see anne-post:21
• Remark 2.3: how to get back to graph norms
• Definition 2.4: $\varepsilon$-separation and $N$-uniform $\eta$-cover
• Proposition 2.5: non-concentrating property for union of balls
• Remark 2.6: on the new constant
• Definition 2.8: canonical energy form on manifold with handles
• Proposition 2.9: modification to a smooth manifold
• Theorem 3.1: fading handles I
• Corollary 3.2: fading handles I