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Arctic Char
Arctic Char
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Let $M$ be a closed Riemannian manifold. AI have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.

(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point of the area functional. Considering the 2nd variation, one has the Jacobi operator. So the kernel of the Jacobi operator $K$ corresponds to the first order deformation of minimal submanifolds near $N$. Will there be a map

$$ K \to \{\text{all minimal submanifolds in }M\}$$

Such that it is locally surjective around $N$?

(2) What can we say about the structure of the set of all minimal submanifolds in $M$?

If the question is too board, please feel free to impose restrictions (for example, dimension on $M$, $N$ or even some specific $M$)

Remark: In some very special case (for example special Lagrangian submanifolds in a Calabi-Yau $M$), $K$ (need to restrict to Lagrangian variation here) actually parametrizes locally all nearby special Lagrangians. But the proof uses the fact that all special Lagrangians are calibrated, so cannot be generalized.

Let $M$ be a closed Riemannian manifold. A have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.

(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point of the area functional. Considering the 2nd variation, one has the Jacobi operator. So the kernel of the Jacobi operator $K$ corresponds to the first order deformation of minimal submanifolds near $N$. Will there be a map

$$ K \to \{\text{all minimal submanifolds in }M\}$$

Such that it is locally surjective around $N$?

(2) What can we say about the structure of the set of all minimal submanifolds in $M$?

If the question is too board, please feel free to impose restrictions (for example, dimension on $M$, $N$ or even some specific $M$)

Remark: In some very special case (for example special Lagrangian submanifolds in a Calabi-Yau $M$), $K$ (need to restrict to Lagrangian variation here) actually parametrizes locally all nearby special Lagrangians. But the proof uses the fact that all special Lagrangians are calibrated, so cannot be generalized.

Let $M$ be a closed Riemannian manifold. I have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.

(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point of the area functional. Considering the 2nd variation, one has the Jacobi operator. So the kernel of the Jacobi operator $K$ corresponds to the first order deformation of minimal submanifolds near $N$. Will there be a map

$$ K \to \{\text{all minimal submanifolds in }M\}$$

Such that it is locally surjective around $N$?

(2) What can we say about the structure of the set of all minimal submanifolds in $M$?

If the question is too board, please feel free to impose restrictions (for example, dimension on $M$, $N$ or even some specific $M$)

Remark: In some very special case (for example special Lagrangian submanifolds in a Calabi-Yau $M$), $K$ (need to restrict to Lagrangian variation here) actually parametrizes locally all nearby special Lagrangians. But the proof uses the fact that all special Lagrangians are calibrated, so cannot be generalized.

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Arctic Char
Arctic Char
  • 534
  • 6
  • 17

A general theory for local moduli space of minimal surface?

Let $M$ be a closed Riemannian manifold. A have several questions concerning the set of all minimal submanifolds (or immersion) in $M$.

(1): Is there a general local theory for the set of minimal submanifold? What I mean is that, let $N \subset M$ be a minimal submanifold, then it corresponds to the critical point of the area functional. Considering the 2nd variation, one has the Jacobi operator. So the kernel of the Jacobi operator $K$ corresponds to the first order deformation of minimal submanifolds near $N$. Will there be a map

$$ K \to \{\text{all minimal submanifolds in }M\}$$

Such that it is locally surjective around $N$?

(2) What can we say about the structure of the set of all minimal submanifolds in $M$?

If the question is too board, please feel free to impose restrictions (for example, dimension on $M$, $N$ or even some specific $M$)

Remark: In some very special case (for example special Lagrangian submanifolds in a Calabi-Yau $M$), $K$ (need to restrict to Lagrangian variation here) actually parametrizes locally all nearby special Lagrangians. But the proof uses the fact that all special Lagrangians are calibrated, so cannot be generalized.