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In this paper, we show an optimal volume growth for self-shrinkers, andestimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator onself-shrinkers, inspired by the first eigenvalue conjecture on minimalhypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates,we can prove a compactness theorem on a class of compact self-shrinkers in$\ir{3}$ obtained by Colding-Minicozzi under weaker conditions.

Volume growth, eigenvalue and compactness for self-shrinkers