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First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin solves the self-dual equations $$ \begin{cases} F_A + [\Phi, \Phi^*]=0\\ d_A^{\prime\prime}\Phi=0 \end{cases} $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a set of solutions to the equation, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a set of solutions to the Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin solves the self-dual equations $$ \begin{cases} F_A + [\Phi, \Phi^*]=0\\ d_A^{\prime\prime}\Phi=0 \end{cases} $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a solution to the equation above, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a solution to this Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

First, let me introduce the background of the problem: While studying chapter four of the Hitchin self-duality equations, I encountered an issue. Hitchin solves the self-dual equations $$ F_A + [\Phi, \Phi^*] $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a set of solutions to the equation, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a set of solutions to the Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin solves the self-dual equations $$ \begin{cases} F_A + [\Phi, \Phi^*]=0\\ d_A^{\prime\prime}\Phi=0 \end{cases} $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a set of solutions to the equation, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a set of solutions to the Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

First, let me introduce the background of the problem: While studying chapter four of the Hitchin self-duality equations, I encountered an issue. Hitchin solves the self-dual equations [ F_A + [\Phi, \Phi^*] ] within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set ((A, \Phi)). My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection (A) is smooth; when (A) is smooth, (\Phi) is naturally smooth. We know that if ((A, \Phi)) is a set of solutions to the equation, and (g) is a unitary gauge transformation, then under the action of (g), ((g^{-1} A g, g^{-1} \Phi g)) is also a set of solutions to the Hitchin equations. So, if (A) is not a smooth connection, we hope to seek a unitary gauge transformation (g) such that (g^{-1} A g) is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every (W^{2,2}) orbit in the (W^{1,2}) space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let ( M ) be a Riemann surface, and ( V ) be a rank 2 complex vector bundle on ( M ). Let ( H ) be a Hermitian metric on ( V ), and denote ( \mathscr{A} ) as the space of smooth unitary connections on ( (V, H) ). Perform a ( W^{1,2} ) Sobolev completion on ( \mathscr{A} ), and denote this as ( \mathscr{A}_1^2 ).

Let ( \mathscr{G} ) be the group of unitary gauge transformations on ( (V, H) ), and ( \mathscr{G}^\mathbb{C} ) be the complexification of ( \mathscr{G} ), essentially the general gauge transformation group on ( V ).

The action of ( \mathscr{G} ) on ( \mathscr{A} ) is given by: [ g \cdot A \mapsto g^{-1} A g ] This action is extended to ( \mathscr{G}^\mathbb{C} ) acting on ( \mathscr{A} ) as follows: [ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g ]

Further, consider the ( W^{2,2} ) completions of ( \mathscr{G} ) and ( \mathscr{G}^\mathbb{C} ), denoted as ( \mathscr{G}_2^2 ) and ( \mathscr{G}^\mathbb{C}_2^2 ), acting on ( \mathscr{A}_1^2 ).

Atiyah and Bott pointed out that in ( \mathscr{A}_1^2 ), on every ( \mathscr{G}^\mathbb{C}_2^2 ) orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ( \mathscr{G}^\mathbb{C}_2^2 ) orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each ( \mathscr{G}_2^2 ) orbit, there exists at least one smooth connection. However, the ( \mathscr{G}_2^2 ) orbits are a lower-dimensional subspace of the ( \mathscr{G}^\mathbb{C}_2^2 ) orbits, thus the problem is non-trivial.

My question is: Does each ( \mathscr{G}_2^2 ) orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

First, let me introduce the background of the problem: While studying chapter four of the Hitchin self-duality equations, I encountered an issue. Hitchin solves the self-dual equations $$ F_A + [\Phi, \Phi^*] $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a set of solutions to the equation, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a set of solutions to the Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?