We would like to show that there exists a smooth metric $g$ and a collection of lines satisfying the Assumptions A1 and A2.

Let $g_{0}$ be an arbitrary smooth metric, flat near singular points. We define germs of lines $l\in{\cal L}_{in}$ in such a way that for each $s\in B^{sing}$ in local coordinates these lines are given by $\{(0,y)|y>0\}$ and $\{(0,y)|y<0\}$. The metric $g$ will coincide with $g_{0}$ in a sufficiently small neighborhood $U=\cup_{s\in B^{sing}}D(s,r_{s})$ of the singular set. Hence Assumption A1 will be satisfied.

In order to construct the whole family of lines we introduce a $3$-dimensional manifold ${\cal M}$ consisting of pairs $(x,P)$ where $x\in B\setminus\overline{U}_{1}$ and $P$ is a half-plane in $T_{x}B$ whose boundary contains zero. Here $U_{1}:=\cup_{s\in B^{sing}}D(s,2r_{s})$ is a larger neighborhood of $B^{sing}$.

We would like to construct a smooth section $v:(x,P)\mapsto v_{(x,P)}\in T_{x}B$ of the pull-back to ${\cal M}$ of the tangent bundle $TB$ satisfying the following conditions:

1. 1.

   for any $(x,P)\in{\cal M}$ one has $v_{(x,P)}\in int(P)\,\,$;

2. 2.

   for any $x\in B\setminus\overline{U}_{1}$ the map $(x,P)\mapsto{{\bf R}}_{>0}^{\times}\cdot v_{(x,P)}$ is an orientation-preserving diffeomorphism

   $$S^{1}\simeq(T_{x}B^{\ast}\setminus\{0\})/{{\bf R}}_{>0}^{\times}\to S^{1}\simeq(T_{x}B\setminus\{0\})/{{\bf R}}_{>0}^{\times}\,\,\,;$$

3. 3.

   for every $l\in{\cal L}_{in}$ there exists a smooth extension of the piece of $l$ in $U_{1}$ to a larger piece intersecting $\partial U_{1}$ such that

   $$\dot{f}_{l}(t)\in{{\bf R}}_{>0}^{\times}\cdot v_{(f_{l}(t),P_{l,t})}\,\,,$$

   for such $t>0$ that $f_{l}(t)\in B\setminus\overline{U}_{1}\,\,$;

Let us associate with the section $v$ a nowhere vanishing vector field $\hat{v}$ on $T^{\ast}(B\setminus\overline{U}_{1})\setminus(Zero\,\,\,Section)$ in the following way:

• •

  For each $(x,\alpha)\in T_{x}^{\ast}B$ the vector $\hat{v}(x,\alpha)$ is tangent to the horizontal distribution associated with the flat connection $\nabla$ (the one which defines the affine structure on $B\setminus B^{sing}$).

• •

  Projection of $\hat{v}(x,\alpha)$ to $B$ coincides with $v_{(x,P_{\alpha})}$, where $P_{\alpha}=\{\gamma|(\alpha,\gamma)>0\}$.

Clearly these conditions determines $\hat{v}$ uniquely. Now we formulate last condition:

1. 4.

   there exits $r_{s}^{\prime}>2r_{s}$ such that for almost all (in the sense of Baire category) initial values $(x_{0},P_{0})\in{\cal M}$ the integral curve of $\hat{v}$ starting at $(x_{0},P_{0})$ reaches the pullback of $B\setminus\cup_{s\in B^{sing}}D(s,r_{s}^{\prime})$ in finite time.

Using the vector field $\hat{v}$ we will construct (under certain genericity assumptions) a set ${\cal L}$ of lines satisfying Assumption A1. Namely, the data consisting of a line $l$ and an integer-valued $1$-form $\alpha_{l}$ (see Section 9) will be an integral line of $\hat{v}$.

We are going to construct lines by induction by the number of collisions. Lines $l\in{\cal L}_{in}$ will be constructed using condition 3. The genericity assumption mentioned after the condition 4 is the assumption that no more than two lines collide and that initial values for newborn lines will be sufficiently generic. Conditions 1 and 4 plus genericity imply that one can parametrize any line $l\in{\cal L}$ by the new “time” $t>0$ such that the Axiom 2 is satisfied. Axiom 6 follows from the condition 2. Other axioms and the Assumption A1 will be satisfied automatically.

Now we would like to discuss Assumption A2.

Proof. In order to satisfy Assumption A2 it suffices to find such $\varepsilon>0$ that for any $x\in B\setminus\overline{U}_{1}$ and any half-plane $P_{x}\subset T_{x}B,\,\,0\in int(P_{x})$ with the distance $dist_{g_{x}}(0,\partial P_{x})=\varepsilon$, and another half-plane $P_{x}^{\prime}\subset T_{x}B$ parallel to $P_{x}$ such that $0\in\partial P_{x}^{\prime}$, one has the following property: if $P_{x+\delta tv_{(x,P_{x}^{\prime})}}$ is the half-plane obtained from $P_{x}$ by a small covariant (with respect to the affine connection $\nabla^{aff}$) shift $\delta t$ in the direction of $v_{(x,P_{x}^{\prime})}$, then

$$dist_{g_{x+\delta tv_{(x,P_{x}^{\prime})}}}(0,P_{x+\delta tv_{(x,P_{x}^{\prime})}})\geq dist_{g_{x}}(0,\partial P_{x})\,\,.$$

Here $g_{x}$ etc. denotes the induced flat metric on the tangent space $T_{x}B$. This property guarantees that the condition $dist_{g_{x}}(0,\partial P_{l,t})\geq\varepsilon$ will propagate along the line. For a new line obtained as a result of collision of $l_{1}$ and $l_{2}$ at the times $t_{1}$ and $t_{2}$ respectively one has

$$dist_{g_{x}}(0,\partial P_{l,0})\geq\min\{dist_{g_{x}}(0,\partial P_{l_{1},t_{1}}),dist_{g_{x}}(0,\partial P_{l_{2},t_{2}})\}$$

since $\partial P_{l,0}$ contains the intersection point $\partial P_{l_{1},t_{1}}\cap\partial P_{l_{2},t_{2}}$, see Figure 7.

One can easily see that the infinitesimal inequality from above is equivalent to

$$\delta tg_{x}(v_{(x,P_{x}^{\prime})},n_{P_{x}^{\prime}})+\varepsilon/2(g_{x+\delta tv_{(x,P_{x}^{\prime})}}-g_{x})(n_{P_{x}^{\prime}},n_{P_{x}^{\prime}})\geq 0$$

(the change of the distance consists of two summands: one corresponds to the shift along $\delta tv_{(x,P_{x})}$ with the fixed metric, and the other one corresponds to the change of the metric). Taking the limit $\delta t\to 0$ we arrive to the inequality for the covariant derivative of the metric with $C=2/\varepsilon$. $\blacksquare$

Now our goal is to construct the field of directions $v$ and the metric $g$ satisfying the conditions 1–4 and the inequality from the last Propostion. This will conclude the construction of the set ${\cal L}$ of lines satisfying the Assumptions A1 and A2.

Since $\partial U_{1}$ is a boundary of the convex set, we can locally model it by the graph of function $y=f(x)$ such that $f^{\prime\prime}(x)>0$, $f^{\prime}(x_{0})=0$. We may assume that $P=P_{0}$ is the upper half-plane. Then we take

$$v_{\left((x,y),P\right)}=\partial/\partial y+{(f(x)-f(x_{0}))/f^{\prime}(x)\over{f(x)-f(x_{0})+f(x)-y}}\,\,\partial/\partial x\,\,.$$

We extend this local model of $v$ near $\partial U_{1}$ to $B\setminus\overline{U}_{1}$ in such a way that conditions 1 and 2 are satisfied. It is clear that we can satisfy conditions 3,4 as well by taking a small perturbation of $v$. On Figure 8 there is a picture of the field $(x,y)\mapsto v_{\left((x,y),P_{0}\right)}$.

For an arbitrary choice of the metric $g$ we have $g(n_{P},v_{z,P})>0$ for all $(z,P)\in{\cal M}$. The problem with inequality

$$(\nabla_{v_{(z,P)}}\,g)(n_{P},n_{P})\leq Cg(n_{P},v_{(z,P)})$$

arises only as the point $z$ approaches $\partial U_{1}$. Indeed, in this case the vector $v_{(z,P)}$ can be very close to the tangent vector to $\partial P_{z}\subset T_{z}B$.

Proof. We need to check that the ratio

$${(\nabla_{v_{(z,P)}}\,g)(n_{P},n_{P})}\over{g(n_{P},v_{(z,P)})}$$

is bounded for $(z,P)\in{\cal M}$.

It suffices to prove the Lemma assuming that $U_{1}$ is the parabolic domain $\{(x,y)\in{{\bf R}}^{2}|y>x^{2}\}$ and $P$ is the upper half-plane. The vector field $v_{(z,P)}$ is given for $z=(x,y)$ by the formulas

$$v_{(z,P)}=\partial/\partial y+{x\over{4x^{2}-2y}}\,\partial/\partial x\,\,.$$

The denominator is equal to $g(n_{P},v_{(z,P)})=\langle dy,v_{(z,P)}\rangle\cdot\sqrt{g(\partial/\partial y,\partial/\partial y)}=\sqrt{g(\partial/\partial y,\partial/\partial y)}=\exp(O(1))$ near $(0,0)$.

The numerator is equal to

$${x\over{4x^{2}-2y}}f_{1}(x,y)+f_{2}(x,y)\,\,,$$

where $f_{1}(x,y)=(\nabla_{\partial/\partial x}\,g)(n_{P},n_{P})$ and $f_{2}(x,y)=(\nabla_{\partial/\partial y}\,g)(n_{P},n_{P})$ are two $C^{\infty}$-functions.

By assumption of the Lemma we have $f_{1}(0,0)=0$. Therefore $|f_{1}(x,y)|\leq const\,\max\{|x|,|w|\}$ where $w=x^{2}-y$ is a convenient local coordinate near the point $(0,0)$. Notice also that $f_{2}(x,y)=O(1)$.

Now we can estimate first summand of the numerator assuming that $|x|$ and $|w|$ are sufficiently small. As we have seen, it is bounded by

$$I:={x\over{x^{2}+w}}O(\max\{|x|,|w|\}\,\,.$$

There are three cases which we need to consider.

a) If $0<w<x^{2}$ then $I={x\over{x^{2}}}O(|x|)=O(1)$.

b) if $x^{2}\leq w<x$ then $I={x\over w}O(|x|)=O(1)$.

c) If $x\leq w\leq 1$ the $I={x\over w}O(|w|)=O(1)$.

We see that the numerator is bounded. This concludes the proof of Lemma. $\blacksquare$

Finally, we have the following result.

Proof: First of all, the condition on $g$ from Lemma 6 is the condition on a loop $g_{|T_{z}B}$ of scalar products on 2-dimensional spaces, here $z\in\partial U_{1}\simeq S^{1}$. We can write $g=\exp(\psi)g_{0}$ where $\det(g_{0})=1$ and $\psi$ is a smooth function. Then we have

$$\nabla_{e_{z}}(\exp{\psi}g_{0})=\exp({\psi})\nabla_{e_{z}}g_{0}+\exp(\psi)\partial_{e_{z}}(\psi)\,g_{0}\,\,.$$

The equation of Lemma 6 gives $\partial_{e_{z}}\psi=-(\nabla_{e_{z}}g_{0})(n_{z},n_{z})/g_{0}(n_{z},n_{z})$. The RHS of this expression is known as long as we know $g_{0}$. Hence we can say that $d\psi=\beta_{g_{0}}$, where $\beta_{g_{0}}$ is a 1-form depending on the restriction $(g_{0})_{|\partial U_{1}}$. We see that it suffices to find such $g_{0}$ that $\int_{\partial U_{1}}\beta_{g_{0}}=0$ (then $\psi$ and hence $g$ does exist).

Let us consider the functional $I(g_{0})=\int_{S^{1}}\beta_{g_{0}}$. We can interpret a metric $g_{0}$ as a point in the Lobachevsky plane ${\cal H}=SL(2,{\bf R})/SO(2)$. More precisely, let us consider the space $S$ of pairs $(g_{0},P)$ where $g_{0}$ is a positive quadratic form on ${\bf R}^{2}$ such that $\det(g_{0})=1$ and $P$ is a half-plane in ${\bf R}^{2}$ (the meaning of $P$ is the inward oriented tangent half-plane to $\partial U_{1}$ at point $z\in\partial U_{1}$). This space is naturally diffeomorphic to $S^{\ast}({\bf R}^{2})\times{\cal H}$. The latter manifold can be identified in $SL(2,{\bf R})$-equivariant way with the manifold consisting of pairs $(x,y)$, where $x\in{\cal H}$ and $y$ belongs to the absolute. Hence $(g_{0})_{|\partial U_{1}}$ is (locally) a non-parametrized path in $S$ (it would be a global path, if the bundle over $S^{1}$ given by the all metrics on $S^{1}$ with the determinant $1$ was trivial).

Next we observe that the variation $\delta I(g_{0})=\int_{N}\omega$, where $N$ is a $2$-dimensional surface bounded by the paths defined by $g_{0}$ and $g_{0}+\delta g_{0}$, and $\omega$ is a canonical $SL(2,{\bf R})$-invariant $2$-form on $S$. One can show that even by a small variation of the path defined by $g_{0}$ we can make $I(g_{0})$ an arbitrary real number. In particular, we can find $g_{0}$ such that $I(g_{0})=0$. This concludes the proof of Lemma 6. $\blacksquare$

Summarizing, we have constructed a set of lines satisfying the Assumptions A1 and A2. This concludes the proof of Theorem 5. Thus we have obtained a solution of the Lifting Problem, which is a $K$-analytic K3 surface.