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arXiv · 2610.24964

Complete self-shrinkers with constant norm of the second fundamental form in \(\RR^4\)

Yu Fu·Dan Yang
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Abstract

Let \(X:M^3\to\RR^4\) be a complete self-shrinker with constant squared norm \(S\) of the second fundamental form. We prove that \(S=0\) or \(S=1\), and that \(X(M)\) is isometric to a hyperplane or a standard product \(\SSS^k(\sqrt{k})\times\RR^{3-k}\), \(1\leq k\leq3\). This proves a conjecture proposed by Cheng and Wei for three-dimensional hypersurfaces.

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Yu Fu, Dan Yang. 2026-10-18. Complete self-shrinkers with constant norm of the second fundamental form in \(\RR^4\). https://arxiv.org/abs/2610.24964

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