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In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ with minimal Maslov number $3$ one can define a Floer theory and hence the Lagrangian will be non-displaceable.

My question is that the definitions for monotone involve constants $\lambda>0,\alpha>0$, one for the definition of $L$ to be monotone and the other for the definition of $M$ to be monotone. My question is that what happens if we ask that $\lambda<0,\alpha<0$ ? Or even in the case $\lambda>0$ and $\alpha<0$ so they have opposite signs.

From From what I've seen I would expect that the theory still works as long as the constants have the same sign but I would love to hear some input on this. Any insight is appreciated, thanks in advance.

In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ with minimal Maslov number $3$ one can define a Floer theory and hence the Lagrangian will be non-displaceable.

My question is that the definitions for monotone involve constants $\lambda>0,\alpha>0$, one for the definition of $L$ to be monotone and the other for the definition of $M$ to be monotone. My question is that what happens if we ask that $\lambda<0,\alpha<0$ ? Or even in the case $\lambda>0$ and $\alpha<0$ so they have opposite signs.

From what I've seen I would expect that the theory still works as long as the constants have the same sign but I would love to hear some input on this. Any insight is appreciated, thanks in advance.

In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ with minimal Maslov number $3$ one can define a Floer theory and hence the Lagrangian will be non-displaceable.

My question is that the definitions for monotone involve constants $\lambda>0,\alpha>0$, one for the definition of $L$ to be monotone and the other for the definition of $M$ to be monotone. My question is that what happens if we ask that $\lambda<0,\alpha<0$ ? From what I've seen I would expect that the theory still works as long as the constants have the same sign but I would love to hear some input on this. Any insight is appreciated, thanks in advance.

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Someone
  • 791
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Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians

In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ with minimal Maslov number $3$ one can define a Floer theory and hence the Lagrangian will be non-displaceable.

My question is that the definitions for monotone involve constants $\lambda>0,\alpha>0$, one for the definition of $L$ to be monotone and the other for the definition of $M$ to be monotone. My question is that what happens if we ask that $\lambda<0,\alpha<0$ ? Or even in the case $\lambda>0$ and $\alpha<0$ so they have opposite signs.

From what I've seen I would expect that the theory still works as long as the constants have the same sign but I would love to hear some input on this. Any insight is appreciated, thanks in advance.