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Definition of $C^1,gamma$-hypersurface

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I'm studying by myself Geometric Measure Theory by the book "Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory" written by Francesco Maggi and he assumes that the reader is familiar with the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$ on the part $textbfIII$ of the book, which deals with the regularity theory and analysis of singularities, but I'm not familiar with this definition. I found something about $mathcalC^1,gamma$-maps with $gamma in (0,1]$ here and here, but nothing about $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$.

Can someone provide me the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$? If it is necessary to know previously some properties of these kind of hypersurfaces before continuing my study in GMT, could anyone recommend to me some reference for study these properties?

Thanks in advance!

differential-geometry reference-request definition geometric-measure-theory

share|cite|improve this question

edited Mar 16 at 12:47

George

asked Mar 15 at 1:51

GeorgeGeorge

850615

$endgroup$

add a comment | 

0

$begingroup$

I'm studying by myself Geometric Measure Theory by the book "Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory" written by Francesco Maggi and he assumes that the reader is familiar with the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$ on the part $textbfIII$ of the book, which deals with the regularity theory and analysis of singularities, but I'm not familiar with this definition. I found something about $mathcalC^1,gamma$-maps with $gamma in (0,1]$ here and here, but nothing about $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$.

Can someone provide me the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$? If it is necessary to know previously some properties of these kind of hypersurfaces before continuing my study in GMT, could anyone recommend to me some reference for study these properties?

Thanks in advance!

differential-geometry reference-request definition geometric-measure-theory

share|cite|improve this question

edited Mar 16 at 12:47

George

asked Mar 15 at 1:51

GeorgeGeorge

850615

$endgroup$

add a comment | 

$begingroup$

I'm studying by myself Geometric Measure Theory by the book "Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory" written by Francesco Maggi and he assumes that the reader is familiar with the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$ on the part $textbfIII$ of the book, which deals with the regularity theory and analysis of singularities, but I'm not familiar with this definition. I found something about $mathcalC^1,gamma$-maps with $gamma in (0,1]$ here and here, but nothing about $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$.

Can someone provide me the definition of $mathcalC^1,gamma$-hypersurface with $gamma in (0,1)$? If it is necessary to know previously some properties of these kind of hypersurfaces before continuing my study in GMT, could anyone recommend to me some reference for study these properties?

Thanks in advance!

differential-geometry reference-request definition geometric-measure-theory

share|cite|improve this question

edited Mar 16 at 12:47

George

asked Mar 15 at 1:51

GeorgeGeorge

850615

$endgroup$

share|cite|improve this question

edited Mar 16 at 12:47

George

asked Mar 15 at 1:51

GeorgeGeorge

850615

share|cite|improve this question

edited Mar 16 at 12:47

George

asked Mar 15 at 1:51

GeorgeGeorge

850615

asked Mar 15 at 1:51

GeorgeGeorge

850615

asked Mar 15 at 1:51

GeorgeGeorge

850615

1 Answer
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2

$begingroup$

I would just take that to mean that the hypersurface is a $(d-1)$-dimensional $C^1,gamma$ manifold.

You could thus think of the hypersurface as being locally given by the graph of a $C^1,gamma$ function (perhaps upon reorienting and/or relabeling your coordinate axes).

Edit: I pulled out my copy of Maggi's book and you can look at Thm 26.3 for a precise statement of this regularity theorem (including, of course, the technical details of what is meant by a hypersurface of class $C^1,gamma$). He additionally includes some estimates on the Lipschitz function(s) used to locally characterize the hypersurface.

share|cite|improve this answer

edited Mar 15 at 14:41

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was thinking that $C^1,gamma$-hypersurfaces are exactly this, but I was not sure. Thanks for the reference of the theorem $26.3$. I didn't realize where in the statement or in the proof of this theorem the author gives the technical details of the definition of $C^1,gamma$-hypersurface, but I didn't start to read part $textbfIII$ of Maggi's book. I will start read this part and try realize the technical details of the definition.
  $endgroup$
  – George
  Mar 16 at 12:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You were spot on. I'm glad I could help. It's always a pleasure when I get to talk math with people.
  $endgroup$
  – Gary Moon
  Mar 16 at 19:18
  

  
  
  
  

add a comment | 

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2

$begingroup$

I would just take that to mean that the hypersurface is a $(d-1)$-dimensional $C^1,gamma$ manifold.

You could thus think of the hypersurface as being locally given by the graph of a $C^1,gamma$ function (perhaps upon reorienting and/or relabeling your coordinate axes).

Edit: I pulled out my copy of Maggi's book and you can look at Thm 26.3 for a precise statement of this regularity theorem (including, of course, the technical details of what is meant by a hypersurface of class $C^1,gamma$). He additionally includes some estimates on the Lipschitz function(s) used to locally characterize the hypersurface.

share|cite|improve this answer

edited Mar 15 at 14:41

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was thinking that $C^1,gamma$-hypersurfaces are exactly this, but I was not sure. Thanks for the reference of the theorem $26.3$. I didn't realize where in the statement or in the proof of this theorem the author gives the technical details of the definition of $C^1,gamma$-hypersurface, but I didn't start to read part $textbfIII$ of Maggi's book. I will start read this part and try realize the technical details of the definition.
  $endgroup$
  – George
  Mar 16 at 12:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You were spot on. I'm glad I could help. It's always a pleasure when I get to talk math with people.
  $endgroup$
  – Gary Moon
  Mar 16 at 19:18
  

  
  
  
  

add a comment | 

2

$begingroup$

I would just take that to mean that the hypersurface is a $(d-1)$-dimensional $C^1,gamma$ manifold.

You could thus think of the hypersurface as being locally given by the graph of a $C^1,gamma$ function (perhaps upon reorienting and/or relabeling your coordinate axes).

Edit: I pulled out my copy of Maggi's book and you can look at Thm 26.3 for a precise statement of this regularity theorem (including, of course, the technical details of what is meant by a hypersurface of class $C^1,gamma$). He additionally includes some estimates on the Lipschitz function(s) used to locally characterize the hypersurface.

share|cite|improve this answer

edited Mar 15 at 14:41

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I was thinking that $C^1,gamma$-hypersurfaces are exactly this, but I was not sure. Thanks for the reference of the theorem $26.3$. I didn't realize where in the statement or in the proof of this theorem the author gives the technical details of the definition of $C^1,gamma$-hypersurface, but I didn't start to read part $textbfIII$ of Maggi's book. I will start read this part and try realize the technical details of the definition.
  $endgroup$
  – George
  Mar 16 at 12:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You were spot on. I'm glad I could help. It's always a pleasure when I get to talk math with people.
  $endgroup$
  – Gary Moon
  Mar 16 at 19:18
  

  
  
  
  

add a comment | 

$begingroup$

I would just take that to mean that the hypersurface is a $(d-1)$-dimensional $C^1,gamma$ manifold.

You could thus think of the hypersurface as being locally given by the graph of a $C^1,gamma$ function (perhaps upon reorienting and/or relabeling your coordinate axes).

Edit: I pulled out my copy of Maggi's book and you can look at Thm 26.3 for a precise statement of this regularity theorem (including, of course, the technical details of what is meant by a hypersurface of class $C^1,gamma$). He additionally includes some estimates on the Lipschitz function(s) used to locally characterize the hypersurface.

share|cite|improve this answer

edited Mar 15 at 14:41

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

$endgroup$

share|cite|improve this answer

edited Mar 15 at 14:41

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

answered Mar 15 at 2:33

Gary MoonGary Moon

66616

answered Mar 15 at 2:33

Gary MoonGary Moon

66616