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Let $Sigma^2 subseteq mathbb{R}^3$ be a complete minimal surface and let's assume that there exists a smooth regular curve $gamma colon I to Sigma$ such that $K(gamma(t)) = 0$ for all $t in I$, where $K$ is the Gauss curvature of $Sigma$.

Can I conclude that $Sigma$ is flat, i.e. a plane?

differential-geometry elliptic-equations minimal-surfaces

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asked Mar 20 at 18:25

Math_touristMath_tourist

764

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  Yes. Do you know how to interpret the Gauss map of a minimal surface as a holomorphic mapping to $Bbb CP^1$?
  $endgroup$
  – Ted Shifrin
  Mar 20 at 19:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Right, but then how do you conclude? Moreover, what about the higher dimensional version of my question? I.e. if $Sigma^n subseteq mathbb{R}^{n+1}$ is a minimal hypersurfaces such that its second fundamental form vanishes along a certain $Gamma^{n-1} subseteq Sigma^n$, can I still conclude that $Sigma$ is flat? In this case I can't use complex analysis argument, right?
  $endgroup$
  – Math_tourist
  Mar 20 at 19:33
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
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  I don't know about the higher-dimensional situation. But with regard to the first question: This is the identity principle for holomorphic functions — on a connected set, if a holomorphic function is zero on a set of points with a limit point, then it is identically zero.
  $endgroup$
  – Ted Shifrin
  Mar 20 at 20:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ah right!!! Thanks a lot!!
  $endgroup$
  – Math_tourist
  Mar 20 at 21:03
  

  
  
  
  

add a comment | 

3

1

$begingroup$

Let $Sigma^2 subseteq mathbb{R}^3$ be a complete minimal surface and let's assume that there exists a smooth regular curve $gamma colon I to Sigma$ such that $K(gamma(t)) = 0$ for all $t in I$, where $K$ is the Gauss curvature of $Sigma$.

Can I conclude that $Sigma$ is flat, i.e. a plane?

differential-geometry elliptic-equations minimal-surfaces

share|cite|improve this question

asked Mar 20 at 18:25

Math_touristMath_tourist

764

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes. Do you know how to interpret the Gauss map of a minimal surface as a holomorphic mapping to $Bbb CP^1$?
  $endgroup$
  – Ted Shifrin
  Mar 20 at 19:07
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Right, but then how do you conclude? Moreover, what about the higher dimensional version of my question? I.e. if $Sigma^n subseteq mathbb{R}^{n+1}$ is a minimal hypersurfaces such that its second fundamental form vanishes along a certain $Gamma^{n-1} subseteq Sigma^n$, can I still conclude that $Sigma$ is flat? In this case I can't use complex analysis argument, right?
  $endgroup$
  – Math_tourist
  Mar 20 at 19:33
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't know about the higher-dimensional situation. But with regard to the first question: This is the identity principle for holomorphic functions — on a connected set, if a holomorphic function is zero on a set of points with a limit point, then it is identically zero.
  $endgroup$
  – Ted Shifrin
  Mar 20 at 20:57
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ah right!!! Thanks a lot!!
  $endgroup$
  – Math_tourist
  Mar 20 at 21:03
  

  
  
  
  

add a comment | 

$begingroup$

Let $Sigma^2 subseteq mathbb{R}^3$ be a complete minimal surface and let's assume that there exists a smooth regular curve $gamma colon I to Sigma$ such that $K(gamma(t)) = 0$ for all $t in I$, where $K$ is the Gauss curvature of $Sigma$.

Can I conclude that $Sigma$ is flat, i.e. a plane?

differential-geometry elliptic-equations minimal-surfaces

share|cite|improve this question

asked Mar 20 at 18:25

Math_touristMath_tourist

764

$endgroup$

share|cite|improve this question

asked Mar 20 at 18:25

Math_touristMath_tourist

764

share|cite|improve this question

asked Mar 20 at 18:25

Math_touristMath_tourist

764

asked Mar 20 at 18:25

Math_touristMath_tourist

764

asked Mar 20 at 18:25

Math_touristMath_tourist

764