Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise it.

Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. The torus is equiped with the standard metric and the corresponding Hodge operator is denoted by $^*$.

Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$ and $|^*\alpha(X)|\leq 1$. The later norm is the usual supnorm of the space of functions. Then $K$ is a convex $C^1$- closed subset of all $C^1$ 1-forms on $\mathbb{T}^2$.

Is $K$ a compact subset of the space of 1-forms with respect to $C^1$ topology? If the answer is affirmative. according to the Krein Millman theorem, what is a presise description of its extreme points of $K$?

Does the topological structure of $K$ depends on chosing the vector field $X$ tangent to our initial Kronecker foliation of torus? Does the topological structure of $K$ depend on the slope of our Kronecker foliation?

Motivation:

A motivation for this question is the following:

In this post and some other related linked posts we try to find a Riemannian metric compatible to orbits of a non vanishing vector fields. Choosing various metrics enable us to have different curvatuare functions. Possessing an appropriate curvature function is very essential for appllying the Gauss Bonnet theorem to the problem of limit cycles of vctor fields.(For counting them as closed geodesics). So this situation leads us to think about the diversity of closed differential 1-forms $\alpha$ with $\alpha(X)=1$. Under these conditions, in particular the propery of closed convexity of this set $K$. one is tempt to be curious about the presice description of possible extrem points of $K$.

Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise it.

Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. The torus is equiped with the standard metric and the corresponding Hodge operator is denoted by $*$.

Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$ and $|*\alpha(X)|\leq 1$. The later norm is the usual supnorm of the space of functions. Then $K$ is a convex $C^1$- closed subset of all $C^1$ 1-forms on $\mathbb{T}^2$.

Is $K$ a compact subset of the space of 1-forms with respect to $C^1$ topology? If the answer is affirmative. according to the Krein Millman theorem, what is a presise description of its extreme points of $K$?

Does the topological structure of $K$ depends on chosing the vector field $X$ tangent to our initial Kronecker foliation of torus? Does the topological structure of $K$ depend on the slope of our Kronecker foliation?

Motivation:

A motivation for this question is the following:

In this post and some other related linked posts we try to find a Riemannian metric compatible to orbits of a non vanishing vector fields. Choosing various metrics enable us to have different curvatuare functions. Possessing an appropriate curvature function is very essential for appllying the Gauss Bonnet theorem to the problem of limit cycles of vctor fields.(For counting them as closed geodesics). So this situation leads us to think about the diversity of closed differential 1-forms $\alpha$ with $\alpha(X)=1$. Under these conditions, in particular the propery of closed convexity of this set $K$. one is tempt to be curious about the presice description of possible extrem points of $K$.