The second gap on complete self-shrinkers

In this paper, we study complete self-shrinkers in Euclidean space and prove that an n n -dimensional self-shrinker alttext="double-struck upper R Superscript n plus 1"> R + 1 encoding="application/x-tex">\mathbb {R}^{n+1} is isometric to either n"> {R}^{n} , alttext="upper S Baseline left-parenthesis StartRoot EndRoot right-parenthesis"> S ( stretchy="false">) encoding="application/x-tex">S^{n}(\sqrt {n}) or k right-parenthesis times double-struck minus k"> k × −<!-- − encoding="application/x-tex">S^k (\sqrt {k})\times \mathbb {R}^{n-k} alttext="1 less-than-or-equal-to ≤<!-- ≤ encoding="application/x-tex">1\leq k\leq n-1 if the squared norm S"> encoding="application/x-tex">S of second fundamental form, alttext="f 3"> f 3 encoding="application/x-tex">f_3 are constant satisfies greater-than 1.83379"> &gt; 1.83379 encoding="application/x-tex">S&gt;1.83379 . We should remark condition polynomial volume growth not assumed.