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Upper volume bounds for submanifolds

2

$begingroup$

Let $f:Mto (N,g)$ be an immersion of a compact smooth manifold $M$ into a compact Riemannian manifold $(N,g)$. Let
$$
mathrm{Vol}(M,f^*g)=int_M dmathrm{vol}_{f^*g}
$$
be the volume of $M$ with respect to the pullback metric $f^*g$.

My question: Can we get an upper bound on the volume of $(M,f^*g)$ in terms of curvature properties of the metric $g$?

One possible idea I had: suppose that $(N,g)$ has Ricci curvature bounded below by $kappa$, i.e. there exists a constant $kappain mathbb{R}$ such that
$$
mathrm{Ric}^Ngeq (n-1)kappa.
$$
Does it follow that $(M,f^*g)$ also has Ricci curvature bounded below by some $kappa'in mathbb{R}$? If so, then we can bound $mathrm{Vol}(M,f^*g)$ above using the Bishop-Gromov inequality.

differential-geometry metric-spaces riemannian-geometry smooth-manifolds

share|cite|improve this question

asked Mar 26 at 1:19

srpsrp

2828

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  First, there is always a $kappa'$, as a consequence of compactness. Second, control of $kappa'$ is not enough - you can embed $S^1$ into $S^3$ with arbitrarily large length, but $kappa' = 0$ on $S^1$. Perhaps control of the second fundamental form is enough?
  $endgroup$
  – Jason DeVito
  Mar 26 at 1:29
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You have to restrict to embedding, or you can wrap around itself as many times as you wish.
  $endgroup$
  – Arctic Char
  Mar 26 at 3:02
  

  
  
  
  

add a comment | 

2

$begingroup$

Let $f:Mto (N,g)$ be an immersion of a compact smooth manifold $M$ into a compact Riemannian manifold $(N,g)$. Let
$$
mathrm{Vol}(M,f^*g)=int_M dmathrm{vol}_{f^*g}
$$
be the volume of $M$ with respect to the pullback metric $f^*g$.

My question: Can we get an upper bound on the volume of $(M,f^*g)$ in terms of curvature properties of the metric $g$?

One possible idea I had: suppose that $(N,g)$ has Ricci curvature bounded below by $kappa$, i.e. there exists a constant $kappain mathbb{R}$ such that
$$
mathrm{Ric}^Ngeq (n-1)kappa.
$$
Does it follow that $(M,f^*g)$ also has Ricci curvature bounded below by some $kappa'in mathbb{R}$? If so, then we can bound $mathrm{Vol}(M,f^*g)$ above using the Bishop-Gromov inequality.

differential-geometry metric-spaces riemannian-geometry smooth-manifolds

share|cite|improve this question

asked Mar 26 at 1:19

srpsrp

2828

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  First, there is always a $kappa'$, as a consequence of compactness. Second, control of $kappa'$ is not enough - you can embed $S^1$ into $S^3$ with arbitrarily large length, but $kappa' = 0$ on $S^1$. Perhaps control of the second fundamental form is enough?
  $endgroup$
  – Jason DeVito
  Mar 26 at 1:29
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You have to restrict to embedding, or you can wrap around itself as many times as you wish.
  $endgroup$
  – Arctic Char
  Mar 26 at 3:02
  

  
  
  
  

add a comment | 

$begingroup$

Let $f:Mto (N,g)$ be an immersion of a compact smooth manifold $M$ into a compact Riemannian manifold $(N,g)$. Let
$$
mathrm{Vol}(M,f^*g)=int_M dmathrm{vol}_{f^*g}
$$
be the volume of $M$ with respect to the pullback metric $f^*g$.

My question: Can we get an upper bound on the volume of $(M,f^*g)$ in terms of curvature properties of the metric $g$?

One possible idea I had: suppose that $(N,g)$ has Ricci curvature bounded below by $kappa$, i.e. there exists a constant $kappain mathbb{R}$ such that
$$
mathrm{Ric}^Ngeq (n-1)kappa.
$$
Does it follow that $(M,f^*g)$ also has Ricci curvature bounded below by some $kappa'in mathbb{R}$? If so, then we can bound $mathrm{Vol}(M,f^*g)$ above using the Bishop-Gromov inequality.

differential-geometry metric-spaces riemannian-geometry smooth-manifolds

share|cite|improve this question

asked Mar 26 at 1:19

srpsrp

2828

$endgroup$

share|cite|improve this question

asked Mar 26 at 1:19

srpsrp

2828

share|cite|improve this question

asked Mar 26 at 1:19

srpsrp

2828

asked Mar 26 at 1:19

srpsrp

2828

asked Mar 26 at 1:19

srpsrp

2828