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Minimally embedded Riemann surfaces in $\mb{S}^3$ and the conformal deformation of their metrics

Santiago R. Simanca

TL;DR

This work develops a framework connecting Willmore-type intrinsic functionals with ambient diffeomorphisms to realize area-preserving conformal deformations of metrics on minimally embedded surfaces in the standard sphere. It proves that such deformations can be implemented by ambient conformal diffeomorphisms preserving minimality of the embedding, and identifies when the Willmore-energy-minimizing representative can be reached within a distinguished conformal class, notably for Lawson equilateral surfaces. Applying the theory to Lawson surfaces, it shows that along any area-preserving conformal deformation within the appropriate class there is a corresponding minimal isometric deformation to a metric of constant scalar curvature, and derives sharp bounds for a sigma-invariant controlling Willmore energy and scalar curvature via $\sigma(\Sigma)$. The results unify Willmore-type variational quantities with conformal geometry of Lawson surfaces and provide a precise extremal framework and equality characterizations across genera.

Abstract

We prove that if $f_g: (Σ,g) \rightarrow (\mb{S}^{2+p},\tg)$ is a smooth minimal isometric embedding of a Riemannian surface $(Σ,g)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area preserving conformal deformations of $g$ on $f_g(Σ)$, then there exists a path of conformal diffeomorphism $F_t: (\mb{S}^{2+p}, F_t^*\tg) \rightarrow (\mb{S}^{2+p},\tg)$ that starts at $\BOne_{\mb{S}^{2+p}}$, set theoretically fixes $f_g(Σ)$ for all $t$, and it is such that $F^{*}_t \tilde{g}\mid_{f_g(M)}=g_t$ with $f_{g_t}: (Σ,g_t) \rightarrow (\mb{S}^{2+p},\tg)$ a path of minimal embedding deformations of the initial $f_g$. We apply this result to the Lawson surface $( Σ,g)=(ξ_{k/m,m}, g_{ξ_{k/m,m}})$, $m|k>1$, to conclude that if $a=μ_{g_{ξ_{k/m,m}}}(Σ)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area $a$ metrics conformal deformations of $g_{k/m,m}$ to a metric $g_a$ of scalar curvature $4πχ(Σ)/a$, then $f_{g_{ξ_{k/m,m}}}: (ξ_{k/m,m},g_{ξ_{k/m,m}}) \rightarrow (\mb{S}^3, \tg)$ has associated minimal isometric conformal deformations $f_{g_t}$ to the isometric embedding $f_{g_a}$ of $g_a$, in sharp contrast with the situation of the standard sphere $ξ_{0,1}$ and Clifford torus $(ξ_{1,1}$, which are the only orientable Riemannian surfaces of genus $0$ and $1$ isometrically embedded into $(\mb{S}^3,\tg)$ as minimal surfaces. If $σ^2(Σ):=\sup_{[g]\in \mc{C}(Σ)}(4πχ( Σ))^2/\left(\frac{1}{4}\inf_{g\in [g]}\mc{W}_{f_g}(Σ)\right)$, $\mc{W}_{f_g}(Σ)$ the Willmore energy of $f_g$ and $\mc{C}( Σ)$ the space of classes, then $(4πχ(Σ))^2/\left( \frac{1} {4}\mc{W}_{f_g}(Σ) \right) \leq σ^2(Σ)=(4πχ( Σ))^2/\left(\frac{1}{4}\mc{W}_{f_{g_{ξ_{k,1}}}}(Σ) \right)$, and we describe the $f_g$s for which the equality is achieved.

Minimally embedded Riemann surfaces in $\mb{S}^3$ and the conformal deformation of their metrics

TL;DR

This work develops a framework connecting Willmore-type intrinsic functionals with ambient diffeomorphisms to realize area-preserving conformal deformations of metrics on minimally embedded surfaces in the standard sphere. It proves that such deformations can be implemented by ambient conformal diffeomorphisms preserving minimality of the embedding, and identifies when the Willmore-energy-minimizing representative can be reached within a distinguished conformal class, notably for Lawson equilateral surfaces. Applying the theory to Lawson surfaces, it shows that along any area-preserving conformal deformation within the appropriate class there is a corresponding minimal isometric deformation to a metric of constant scalar curvature, and derives sharp bounds for a sigma-invariant controlling Willmore energy and scalar curvature via . The results unify Willmore-type variational quantities with conformal geometry of Lawson surfaces and provide a precise extremal framework and equality characterizations across genera.

Abstract

We prove that if is a smooth minimal isometric embedding of a Riemannian surface , and is a path of area preserving conformal deformations of on , then there exists a path of conformal diffeomorphism that starts at , set theoretically fixes for all , and it is such that with a path of minimal embedding deformations of the initial . We apply this result to the Lawson surface , , to conclude that if , and is a path of area metrics conformal deformations of to a metric of scalar curvature , then has associated minimal isometric conformal deformations to the isometric embedding of , in sharp contrast with the situation of the standard sphere and Clifford torus , which are the only orientable Riemannian surfaces of genus and isometrically embedded into as minimal surfaces. If , the Willmore energy of and the space of classes, then , and we describe the s for which the equality is achieved.

Paper Structure

This paper contains 5 sections, 4 theorems, 32 equations.

Table of Contents

  1. Isometric embeddings into the standard sphere
  2. Principal example: Lawson's minimal surfaces immersed into ${\mathbb S}^3$
  3. Conformal deformations of isometric embeddedings of surfaces
  4. Minimal embeddings of surfaces
  5. The notion of $\sigma$ invariant for surfaces

Key Result

Theorem 1

Suppose that $f_g: (\Sigma^k,g) \rightarrow ({\mathbb S}^{2+q},\tilde{g})$ is a minimal isometric embedding of a closed surface $(\Sigma^k,g)$, and that $[0,1]\ni t \rightarrow g_t=e^{2u(t)}g$ is a path of area preserving conformal deformations of $g$ on $f_{g}(\Sigma)$. Then there exists a path of

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 4
  • Theorem 5