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Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not uniformly Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen. 1990.) https://link.springer.com/article/10.1007/BF01453583

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not uniformly Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen, 1990, MR1032941, Zbl 0733.32008).

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not uniformly Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen. 1990.)

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not uniformly Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen. 1990.) https://link.springer.com/article/10.1007/BF01453583

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen. 1990.)

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\omega$ for some $\epsilon>0$. This implies that $X$ is a Stein manifold.

1. Are there obstructions on $(X,\omega)$ for $X$ to be uniformly Stein with respect to $\omega$ ?

2. Are there examples of Stein manifolds which are not uniformly Stein with respect to some Kähler metric ?

I am mostly interested in the case where $X$ is the universal cover of a compact Kähler manifold $X/\Gamma$ and the Kähler metric is the pullback $\pi^*\omega$ of some Kähler metric $\omega$ on $X/\Gamma$. Being uniformly Stein is then intrinsic, in the sense that it does not depend on the choice of the Kähler metric $\omega$ on $X/\Gamma$.

3. Are there examples of compact Kähler manifolds whose universal cover is Stein but not uniformly Stein ?

For universal covers of compact manifolds, it is enough to produce a map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\pi^*\omega$, because of the existence of an exhaustion whose Levi form is bounded from below (T. Napier. Convexity properties of coverings of smooth projective varieties. Mathematische Annalen. 1990.)