Return to Question

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Any help is appreciated. Thanks in advance.

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Also does aynone know a reference where I can see examples of $J$-holomorphic disks $w:(D^2,\partial D^2)\rightarrow (\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$? And if there is some sort of classification for them.

Any help is appreciated. Thanks in advance.

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Also does anyone know a sequence $u_{\alpha}\in \mathcal{M}_{J,\phi}(x,x)$ such that it coverges to a disk bubble ?

Any help is appreciated. Thanks in advance.

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Any help is appreciated. Thanks in advance.

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Any help is appreciated. Thanks in advance.

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the compactness properties of the moduli spaces $\mathcal{M}_{J}(x,y)$. If we take a sequence $u_{\alpha}\in \mathcal{M}_{J}(x,y)$ with constant index $I$ , we will have that bubbling of spheres or disks could occur , and we have the following formula for $k$-cusp trajectories $(\bar u, \bar v, \bar w)$, where $\bar u$ correspond to the broken trajectories , $\bar v$ the bubbling of spheres and $\bar w$ the bubbling of disks:

$I=\sum_{i}Index (u_i)+2\sum_j c_1(v_j)+\sum_l \mu(w_l)$

Then one can use this formula and the fact that the lagrangian submanifolds are monotone to prove compactness properties of the $0$ and $1$ dimensional components of the moduli space.

Now my question , suppose we are in the case $(S^2,S^1)$, where we know that the minimal maslov number $\Sigma_{S^1}=2$, and we are looking at the compactification of the $1$-dimensional components of $\hat {\mathcal{M}}_{J}(x,y)$ for $y\neq x$. And so let's take a sequence $u_{\alpha }\in \hat {\mathcal{M}}_{J}(x,y)$ with constant index $2$. My question is why couldn't we have the bubbling of a single disk occur in this case ? Is it because of the topology that we have in the space of $k$-cusp trajectories , and so when $x\neq z$ there will always have to be broken trajectories ?

Also does anyone know a sequence $u_{\alpha}\in \mathcal{M}_{J,\phi}(x,x)$ such that it coverges to a disk bubble ?

Any help is appreciated. Thanks in advance.