§ 2.31 Sun–Wang–Zhou shrinkers of arbitrarily large entropy879F42A66D65

(Sun–Wang–Zhou Gaussian-width shrinkers) For every $2\leq n\leq 6$ there is a sequence of smooth, connected, embedded self-shrinking hypersurfaces $\Sigma _ k^ n\subset \mathbb R^{n+1}$. They are normalized by $\vec H=-X^\perp /2$, with associated smooth flows $M_ t=\sqrt{-t}\, \Sigma _ k$ for $t<0$. The construction is variational and does not give explicit parametrizations.

• Each shrinker is complete, proper, boundaryless, and has finite entropy. The construction is a multiplicity-one limit of free-boundary min-max hypersurfaces in exhausting balls with metrics approaching the Gaussian metric. The final shrinker has no free boundary.

• Write $G_ n=(4\pi )^{-1}e^{-|x|^2/(2n)}g_{\mathrm{Euc}}$ and let $\omega _ k(\mathbb R^{n+1},G_ n)$ be its Gaussian $k$-width, defined by exhaustion. Then

  \[ F_{0,1}(\Sigma _ k)=\lambda (\Sigma _ k) =\omega _ k(\mathbb R^{n+1},G_ n),\qquad \lambda (\Sigma _ k)\asymp k^{1/(n+1)}. \]

  The growth statement means comparison up to positive constants depending on $n$, and in particular $\lambda (\Sigma _ k)\to \infty $. It is not a closed formula for an individual width or a plotted list of computed values. The unbounded entropies ensure infinitely many different examples, without requiring consecutive widths to be strictly different.

• The Gaussian Morse index for compactly supported normal variations satisfies $\mathrm{index}(\Sigma _ k)\leq k$. This is an upper bound, not an equality or an entropy-stability claim.

• The theorem does not specify a genus, a number of ends, end asymptotics, compactness, or a symmetry group for each member. In dimension two the genus is therefore left unrecorded in the property browser, rather than inferred from the min-max parameter.

• In every dimension $n\geq 2$ and for every $\Lambda >0$, there is a smooth embedded shrinker with $\Lambda <\lambda (\Sigma )<\infty $ and finite Gaussian index. For $n\geq 7$ this follows by taking a lower-dimensional example and a Euclidean product. For example, $\Sigma ^6\times \mathbb R^{n-6}\subset \mathbb R^{n+1}$ has the same entropy as $\Sigma ^6$, and Theorem 5.3 preserves finiteness of index under each line product. These higher-dimensional product examples are noncompact; this extension does not assert the $n$-dimensional Gaussian-width growth law above beyond $n=6$.

Sources: [SunWangZhou2024], Theorem 1.6 and Corollary 1.7, arXiv version 1, PDF p. 4; Definition 5.1 and the proof of Theorem 1.6, pp. 39–42; Definition 5.2, p. 40; Theorem 5.3 and its proof, pp. 42–44. The illustration is a diagram of the width-growth conclusion, not numerical entropy data or a surface model.