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Reference request - Measure of sphere in Riemannian manifold

The space of Riemannian metrics on a given manifold.The measure of a special set in Riemannian manifoldRiemannian measure and Hausdorff measure in a general Riemannian ManifoldDouble of Riemannian manifold.The cut-off function on Riemannian manifoldmeasure on non-oriented Riemannian manifoldIs there a natural Riemannian metric on the space of smooth immersions into a Riemannian manifold?Isometries of Riemannian manifoldMetric on a riemannian manifold with for a constant positive curvatureIs the inverse of the exponential map smooth everywhere on a Complete Riemannian manifold

2

1

$begingroup$

Consider a smooth $2$-dimensional closed Riemannian manifold $M$ and $x in M$. Let $ 0 < r< textinj(M) $ and let $mathcalH^1$ be the Hausdorff measure induced by the Riemannian distance. Where can I find a proof of the fact that there exists a positive constant $C$ s.t.
$$ mathcalH^1(partial B(x,r)) le Cr $$
?

Thank you.

differential-geometry reference-request riemannian-geometry

share|cite|improve this question

asked Mar 11 at 21:21

Bremen000Bremen000

468310

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is not true. You can just consider the upper half plane with its usual hyperbolic metric.
  $endgroup$
  – Frieder Jäckel
  Mar 12 at 18:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please expand your comment? Is the upper half plane a closed Riemannian manifold?
  $endgroup$
  – Bremen000
  Mar 12 at 19:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If your manifold is closed, then your statement is vacuous for big values of $r$. So if the upper half-plane provides a counterexample for small $r$, and you insist on having a closed counterexample, you could take any closed quotient of the upper half plane (ie a hyperbolic surface). Right?
  $endgroup$
  – Seub
  2 days ago
  

  
  
  
  

add a comment | 

2

1

$begingroup$

Consider a smooth $2$-dimensional closed Riemannian manifold $M$ and $x in M$. Let $ 0 < r< textinj(M) $ and let $mathcalH^1$ be the Hausdorff measure induced by the Riemannian distance. Where can I find a proof of the fact that there exists a positive constant $C$ s.t.
$$ mathcalH^1(partial B(x,r)) le Cr $$
?

Thank you.

differential-geometry reference-request riemannian-geometry

share|cite|improve this question

asked Mar 11 at 21:21

Bremen000Bremen000

468310

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is not true. You can just consider the upper half plane with its usual hyperbolic metric.
  $endgroup$
  – Frieder Jäckel
  Mar 12 at 18:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please expand your comment? Is the upper half plane a closed Riemannian manifold?
  $endgroup$
  – Bremen000
  Mar 12 at 19:48
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If your manifold is closed, then your statement is vacuous for big values of $r$. So if the upper half-plane provides a counterexample for small $r$, and you insist on having a closed counterexample, you could take any closed quotient of the upper half plane (ie a hyperbolic surface). Right?
  $endgroup$
  – Seub
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Consider a smooth $2$-dimensional closed Riemannian manifold $M$ and $x in M$. Let $ 0 < r< textinj(M) $ and let $mathcalH^1$ be the Hausdorff measure induced by the Riemannian distance. Where can I find a proof of the fact that there exists a positive constant $C$ s.t.
$$ mathcalH^1(partial B(x,r)) le Cr $$
?

Thank you.

differential-geometry reference-request riemannian-geometry

share|cite|improve this question

asked Mar 11 at 21:21

Bremen000Bremen000

468310

$endgroup$

share|cite|improve this question

asked Mar 11 at 21:21

Bremen000Bremen000

468310

share|cite|improve this question

asked Mar 11 at 21:21

Bremen000Bremen000

468310

asked Mar 11 at 21:21

Bremen000Bremen000

468310

asked Mar 11 at 21:21

Bremen000Bremen000

468310