A Ricci flow ( M , g ( t )) on an n -dimensional Riemannian manifold M is an intrinsic geometric flow. A family of smoothly embedded submanifolds $$(S(t), g_E)$$ ( S ( t ) , g E ) of a fixed Euclidean space $$E = \mathbb {R}^{n+k}$$ E = R n + k is called an extrinsic representation in $$\mathbb {R}^{n+k}$$ R n + k of ( M , g ( t )) if there exists a smooth one-parameter family of isometries $$(S(t), g_E) \rightarrow (M, g(t))$$ ( S ( t ) , g E ) → ( M , g ( t ) ) . When does such a representation exist? We formulate a new way of framing this question for Ricci flows on surfaces of revolution immersed in $$\mathbb {R}^3$$ R 3 . This framework allows us to construct extrinsic representations for the Ricci flow initialized by any compact surface of revolution immersed in $$\mathbb {R}^3$$ R 3 . In particular, we exhibit the first explicit extrinsic representations in $$\mathbb {R}^4$$ R 4 of the Ricci flows initialized by toroidal surfaces of revolution immersed in $$\mathbb {R}^3$$ R 3 .

中文翻译：

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旋转表面上的 Ricci 流：外在观点

n 维黎曼流形 M 上的 Ricci 流 ( M , g ( t )) 是固有几何流。固定欧几里得空间 $$E = \mathbb {R}^{n+k}$$ E 的平滑嵌入子流形 $$(S(t), g_E)$$ ( S ( t ) , g E ) = R n + k 被称为 $$\mathbb {R}^{n+k}$$ R n + k of ( M , g ( t )) 如果存在平滑的单参数等距族$$(S(t), g_E) \rightarrow (M, g(t))$$ ( S ( t ) , g E ) → ( M , g ( t ) ) 。这种表示什么时候存在？对于浸入 $$\mathbb {R}^3$$ R 3 的旋转表面上的 Ricci 流，我们制定了一种新的方法来解决这个问题。该框架允许我们为 Ricci 流构建外部表示，该流由任何浸入 $$\mathbb {R}^3$$ R 3 的紧凑旋转表面初始化。特别是，