Regularity of Geodesics in the Spaces of Convex and Plurisubharmonic Functions

Theorem 0.1. Let U be strictly convex domain, $\varphi_0$ , $\varphi_1$ be two functions from S. Then the weak geodesic u is $C^{1,1}$ regular away from corner points $\partial U \times \{0\} \cup \partial U \times \{1\}$ . For any neighborhood W of $\partial U \times \{0,1\}$ such that $U \times [0,1] \setminus W$ is convex there is a constant $C_W$ dependent on W, U and $\varphi_0$ , $\varphi_1$ such that $$||u||_{C^{1,1}(\overline{U\times[0,1]}\setminus W)} \le C_W.$$ We remark that the above result is stronger than merely local regularity, as it shows that potential blow-up may occur only at the corners. Analogous regularity for weak plurisubharmonic geodesics is not known (see [Abj19] for partial results in the case of $\Omega$ being an Euclidean ball). We can however show the corresponding result for weak geodesics in $\mathcal{H}$ joining two toric plurisubharmonic functions. Recall that a domain $\Omega \subset \mathbb{C}^n$ is Reinhardt if it is invariant with respect to the standard n-dimensional torus action on the coordinates. The axis set is simply $$M := \{ z \in \Omega \mid \exists i \in \{1, \dots, n\} : z_i = 0 \}.$$ <span id="page-3-1"></span>Corollary 0.2. Let $\Omega$ be a smoothly bounded strictly pseudoconvex Reinhardt domain. Suppose that $\phi$ is a weak geodesic solving the problem (0.2). If $\varphi_1, \varphi_2 \in \mathcal{H}$ are toric in the space variables i.e. for all $z, z' \in \Omega$ satisfying |z| = |z'| one has $\varphi_j(z) = \varphi_j(z')$ , j = 1, 2 then $\phi$ is $C^{1,1}$ away from the corner $\partial\Omega \times \partial A$ and $M \times A$ . Given these results and the theory in the Kähler case it is natural to ask whether global $C^{1,1}$ bounds could be obtained. A bit surprisingly we show (see Examples 3.1 and 4.1) that this is not the case: there exist pairs of points in S and H such that the weak geodesics joining them are not globally $C^{1,1}$ . With Example 3.1 in mind it is natural to ask what are the exact conditions guaranteeing that two points $\varphi, \psi \in \mathcal{S}$ can be joined by a smooth geodesic (i.e. problem (0.3) admits a solution u which is smooth in space and time and is furthermore strictly convex up to the boundary for a fixed time). Exploiting the ideas of Li and Wang from [LW15] we can get an exact answer- our second main result in this note:

Theorem 0.3. Let $\varphi, \psi \in \mathcal{S}$ . Then $\varphi$ and $\psi$ can be joined by smooth geodesic if and only if the gradient image of $\varphi$ $$\partial \varphi(U) := \{ p \in \mathbb{R}^n | \exists x \in U : Du(x) = p \}$$ equals the gradient image of $\psi$ . We remark that Theorem 0.3 shares some similarities with Guan's theorem on existence of smooth geodesics in the case of toric compact Kähler manifolds- see [G99]. Indeed, smooth geodesics always exist in the toric setting, but also the image of the moment maps (the analogue of the gradient image) is fixed- it is equal to the Delzant polytope of the toric manifold. We also remak that the problems of finding criteria for existence of smooth geodesics both in the general Kähler and plurisubharmonic setting are widely open. This paper is organized as follows. In Section 1 we recall some preliminary results. We prove Theorem 0.1 and Corollary 0.2 in the next section. In Section 3 we present an example in which the regularity at the corner points fails to be $C^{1,1}$ . In Section 4 a complex analogue of such an example is constructed. In the last section we prove Theorem 0.3. Acknowledgements. Both Authors were supported by Polish National Science Centre grant 2017/26/E/ST1/00955.

Proposition 1.1. Let U be a smoothly bounded strictly convex domain, and $\varphi_0, \varphi_0 \in \mathcal{S}$ . Then the envelope u satisfies the following proprieties: - i) $\det(D_{x,t}^2 u) = 0$ in $U \times (0,1)$ . - ii) $u = \phi$ on $\partial(U \times (0,1))$ . - iii) $|Du|_{(U\times[0,1])} \le C$ . Next lemma is borrowed from [Wan95]. It gives a sufficient condition to glue two convex functions.

Lemma 1.2. Let $U_1, U_2$ be two domains in $\mathbb{R}^n$ with disjoint interiors. Suppose that the convex functions $u_i$ solve $det(D^2u_i) = 0$ in $U_i$ respectively. Suppose that $u_1 = u_2$ and $Du_1 = Du_2$ on $\partial U_1 \cap \partial U_2$ . Then the function $$u(x) \begin{cases} u_1(x), & x \in U_1; \\ u_2(x), & x \in U_2; \\ u_1(x) = u_2(x), & x \in \partial U_1 \cap \partial U_2 \end{cases}$$ is convex in the interior of $\overline{U_1 \cup U_2}$ and solves $det(D^2u) = 0$ there. In the proof of the $C^{1,1}$ regularity we shall need the basic facts from Section 1 in [CNS86]. Recall that a domain $V \in \mathbb{R}^m, m > 1$ satisfies the truncated cone condition if the following holds: there exist constants $\varepsilon, \delta > 0$ such that for every $y \in V$ there is a truncated cone $$K(y) := \{x \neq y | |x - y| < \varepsilon, \text{ and the angle between } x - y \text{ and some unit vector is less than } \delta\}$$ which is contained in V. For our purposes it is sufficient that the domain $U \times (0,1)$ is bounded and convex and thus satisfies the truncated cone condition. The following two results are contained in the aforementioned Section 1 in [CNS86]:

Lemma 1.3. Let V be a convex domain satisfying the truncated cone condition. Given any convex, uniformly Lipschitz function v satisfying for some constant C > 0 and all $x, y \in V$ the bound <span id="page-5-0"></span> $$|v(x) - v(y) - (x - y).Dv(y)| \le C|x - y|^2 \tag{1.1}$$ one has $$|Dv(x) - Dv(y)| \le B|x - y|$$ for some constant B dependent only on V and C. In particular such a function is $C^{1,1}$ regular. We sketch the main idea for the sake of completeness.

Lemma 1.4. Suppose that v is a differentiable uniformly Lipschitz convex function in a convex domain V, such that for every $x_0 \in V$ there is a constant $\epsilon(x_0) > 0$ such that for any $x \in V$ , $|x - x_0| \le \epsilon(x_0)$ one has <span id="page-5-1"></span> $$|v(x) - v(x_0) - (x - x_0).Dv(x_0)| \le C|x - x_0|^2$$ (1.2) for some uniform constant C > 0. Then inequality (1.1) holds with the same constant C. In particular v is $C^{1,1}$ regular.

Lemma 1.5. Let $\Omega$ be a Reinhardt domain in $\mathbb{C}^n$ and u be a bounded plurisubharmonic function on $\Omega$ , invariant with respect to the toric action. Then: (1) The image U of the mapping $$\Omega \ni z \to (log|z_1|, \cdots, log|z_n|) \in \mathbb{R}^n$$ is a domain in $\mathbb{R}^n$ . $\Omega$ is pseudoconvex if and only if U is convex. - (2) The function $v(x) := u(e^{x_1}, \dots, e^{x_n})$ is a convex function in U. Reversely for any bounded convex function v on U the function u defined through this formula extends to a toric plurisubharmonic function on $\Omega$ . - (3) $(dd^c u)^n = 0$ if and only if $det(D^2 v) = 0$ the equivalence continues to hold for bounded singular u and v and then the equalities are understood in weak sense of measures- see [Gut02, Kol05]. In the proof of Corollary 0.2 we shall need some basic geometric facts regarding unbounded convex domains. Recall that if $\overline{U}$ is the closure of such an unbounded convex domain the characteristic cone for the point $x \in \overline{U}$ $$\Gamma := \{ r \in \mathbb{R}^n \mid \forall t \in [0, \infty) \ x + tr \in \overline{U}, \}$$ is nonempty and independent of the base point x. The following lemma says that very long line segments in $\overline{U}$ with one end point in a fixed compact region must be almost parallel to a direction from the characteristic cone:

Lemma 1.6. Let U be an unbounded convex domain and let V be a compact subset of $\overline{U}$ . Then for every $\varepsilon > 0$ there is a positive constant h, dependent on U, V and $\varepsilon$ so that if $x^0 \in V, x^1 \in \overline{U}$ and the length of $[x^0, x^1]$ is more than h then there exists a vector $r \in \Gamma$ such that the angle between r and $[x^0, x^1]$ is less than $\varepsilon$ .

Lemma 1.7. Let U be a convex domain in $\mathbb{R}^n$ (possibly unbounded). Let u be a convex solution to the problem $$\begin{cases} \det(D_{x,t}^{2}u) = 0 \text{ in } U \times (0,1); \\ u = \varphi \text{ in } U \times \{0\}; \\ u = \psi \text{ in } U \times \{1\}; \\ u = 0 \text{ in } \partial U \times (0,1), \end{cases}$$ (1.3) with $\varphi, \psi \in \mathcal{S}$ . Suppose that for any $(x,t) \in U \times (0,1)$ there is a unique line segment $L = [(\xi,0),(\eta,1)]$ containing (x,t) and $\xi, \eta \in U$ , so that u is linear along L. Then $\xi = \xi(x,t)$ and $\eta = \eta(x,t)$ are smooth functions in $\overline{U} \times (0,1)$ .

Lemma 2.1. Let $(x^0, t^0)$ be any point in $U \times (0, 1)$ . Subtracting a linear function if necessary, we may suppose that $$u \ge 0$$ , $u(x^0, t^0) = 0$ . Then $(x^0, t^0)$ is in the convex hull of (n+1) points (not necessarily distinct) $(x^1, t^1), (x^2, t^2), ...(x^{n+1}, t^{n+1})$ in $\partial(U \times (0, 1))$ with $u(x^i, t^i) = 0$ for all $i \in \{1, 2, ... n + 1\}$ .