Skip to main content
MathOverflow

Return to Answer

added 699 characters in body
Source Link
Otis Chodosh
Otis Chodosh
  • 7.2k
  • 3
  • 25
  • 56

On one hand, the answer to the question

Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the Y or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singularities actually appears.

Is that, yes there are "minimal surfaces" in an appropriately generalized sense that have an e.g. "$Y$ times a line" singularity. The easiest example is just the $Y$ times a line! This is a minimal surface in some generalized sense.

On the other hand, the $Y$ times a line cannot arise as the limit of smooth minimal surfaces, under some reasonable assumptions! This is a consequence of the following remarkable work of Brian White: https://projecteuclid.org/euclid.dmj/1240432190. The paper might be very difficult to read, depending on your background in geometric measure theory, but the upshot is: in some loose sense, a "weak" limit of minimal surfaces must have a "mod 2" multiplicity/orientation. It's easy to see that a $Y$ times a line $L$ cannot be given such an orientation so that they cancel out at the spine $\{0\}\times L$.

On the other hand, a "cross" times a line can be given such an orientation (give each of the four half-planes the multiplicity $1$, then this cancels out at the spine. So, a natural question is, can this cross arise as a limit of smooth minimal surfaces? The answer is yes: a rescaling of Scherk's surface http://en.wikipedia.org/wiki/Scherk_surface#Scherk.27s_second_surface will do exactly this.

EDIT: One comment, based on your wording of the question I wanted to clarify one point. I'm not well versed in the theory of almost minimal sets, but it seems that Jean Taylor's theorem is not about "surfaces" but rather "sets." (see, e.g. http://www.math.u-psud.fr/~gdavid/Montreal011.pdf). So, it is not really concerned with minimal surfaces or "almost minimal surfaces." I'll remark that White's theorem that I linked above would apply to a sequence of minimal surfaces with (1) mean curvature uniformly bounded in $L^1$ and boundary length uniformly bounded. So, this seems to suggest that Taylor's theorem is really about a different sort of objects than classical "surfaces."

On one hand, the answer to the question

Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the Y or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singularities actually appears.

Is that, yes there are "minimal surfaces" in an appropriately generalized sense that have an e.g. "$Y$ times a line" singularity. The easiest example is just the $Y$ times a line! This is a minimal surface in some generalized sense.

On the other hand, the $Y$ times a line cannot arise as the limit of smooth minimal surfaces, under some reasonable assumptions! This is a consequence of the following remarkable work of Brian White: https://projecteuclid.org/euclid.dmj/1240432190. The paper might be very difficult to read, depending on your background in geometric measure theory, but the upshot is: in some loose sense, a "weak" limit of minimal surfaces must have a "mod 2" multiplicity/orientation. It's easy to see that a $Y$ times a line $L$ cannot be given such an orientation so that they cancel out at the spine $\{0\}\times L$.

On the other hand, a "cross" times a line can be given such an orientation (give each of the four half-planes the multiplicity $1$, then this cancels out at the spine. So, a natural question is, can this cross arise as a limit of smooth minimal surfaces? The answer is yes: a rescaling of Scherk's surface http://en.wikipedia.org/wiki/Scherk_surface#Scherk.27s_second_surface will do exactly this.

On one hand, the answer to the question

Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the Y or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singularities actually appears.

Is that, yes there are "minimal surfaces" in an appropriately generalized sense that have an e.g. "$Y$ times a line" singularity. The easiest example is just the $Y$ times a line! This is a minimal surface in some generalized sense.

On the other hand, the $Y$ times a line cannot arise as the limit of smooth minimal surfaces, under some reasonable assumptions! This is a consequence of the following remarkable work of Brian White: https://projecteuclid.org/euclid.dmj/1240432190. The paper might be very difficult to read, depending on your background in geometric measure theory, but the upshot is: in some loose sense, a "weak" limit of minimal surfaces must have a "mod 2" multiplicity/orientation. It's easy to see that a $Y$ times a line $L$ cannot be given such an orientation so that they cancel out at the spine $\{0\}\times L$.

On the other hand, a "cross" times a line can be given such an orientation (give each of the four half-planes the multiplicity $1$, then this cancels out at the spine. So, a natural question is, can this cross arise as a limit of smooth minimal surfaces? The answer is yes: a rescaling of Scherk's surface http://en.wikipedia.org/wiki/Scherk_surface#Scherk.27s_second_surface will do exactly this.

EDIT: One comment, based on your wording of the question I wanted to clarify one point. I'm not well versed in the theory of almost minimal sets, but it seems that Jean Taylor's theorem is not about "surfaces" but rather "sets." (see, e.g. http://www.math.u-psud.fr/~gdavid/Montreal011.pdf). So, it is not really concerned with minimal surfaces or "almost minimal surfaces." I'll remark that White's theorem that I linked above would apply to a sequence of minimal surfaces with (1) mean curvature uniformly bounded in $L^1$ and boundary length uniformly bounded. So, this seems to suggest that Taylor's theorem is really about a different sort of objects than classical "surfaces."

Source Link
Otis Chodosh
Otis Chodosh
  • 7.2k
  • 3
  • 25
  • 56

On one hand, the answer to the question

Is there an analogous of this for minimal surfaces (mean curvature = 0)? I know that minimal surfaces are smooth but, are there examples where they kind of have the Y or the tetrahedron singularity?. It is easy to see in experiments that in real soap bubbles this singularities actually appears.

Is that, yes there are "minimal surfaces" in an appropriately generalized sense that have an e.g. "$Y$ times a line" singularity. The easiest example is just the $Y$ times a line! This is a minimal surface in some generalized sense.

On the other hand, the $Y$ times a line cannot arise as the limit of smooth minimal surfaces, under some reasonable assumptions! This is a consequence of the following remarkable work of Brian White: https://projecteuclid.org/euclid.dmj/1240432190. The paper might be very difficult to read, depending on your background in geometric measure theory, but the upshot is: in some loose sense, a "weak" limit of minimal surfaces must have a "mod 2" multiplicity/orientation. It's easy to see that a $Y$ times a line $L$ cannot be given such an orientation so that they cancel out at the spine $\{0\}\times L$.

On the other hand, a "cross" times a line can be given such an orientation (give each of the four half-planes the multiplicity $1$, then this cancels out at the spine. So, a natural question is, can this cross arise as a limit of smooth minimal surfaces? The answer is yes: a rescaling of Scherk's surface http://en.wikipedia.org/wiki/Scherk_surface#Scherk.27s_second_surface will do exactly this.