Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into $CAT(0) $ metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on $CAT(0)$ spaces and extended Crandall-Liggett’s theory of gradient flows from Banach spaces to $CAT(0) $ spaces to obtain the weak solutions for the harmonic map heat flow into $CAT(0)$ spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question to ask if the weak solutions possess the Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and $1\over 2$-Hölder continuous in time, for a wide class of $CAT(0)$ spaces. In the present paper, we give a complete answer to the question. We show that every weak solution of the harmonic map heat flow into $CAT(0)$ spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.

💡 Research Summary

The paper addresses a long‑standing regularity problem for the harmonic map heat flow (HMHF) when the target space is a general CAT(0) metric space, possibly non‑compact and not embeddable into Euclidean space. Classical results by Eells‑Sampson (1964) guarantee smooth long‑time existence and convergence for HMHF between smooth manifolds with non‑positive curvature. Gromov and Schoen (1992) extended the notion of harmonic maps to CAT(0) spaces, prompting the study of the corresponding heat flow. Mayer and Jost (1990s) later constructed weak (semigroup) solutions for HMHF into CAT(0) spaces via the Crandall‑Liggett scheme, establishing existence, uniqueness, and a ½‑Hölder continuity in time, but spatial Lipschitz regularity remained open.

Recent work by Lin, Segatti, Sire, and Wang (2023) proved spatial Lipschitz regularity for a restricted class of CAT(0) spaces that can be realized as subsets of Euclidean space and satisfy a small‑scale almost‑isometry condition. They conjectured that the result should hold for arbitrary CAT(0) spaces, but the lack of a direct link between the Crandall‑Liggett construction and higher regularity made the problem difficult.

The present paper gives a complete affirmative answer. The main theorem (Theorem 1.3) states that for any complete Riemannian manifold (M) with Ricci curvature bounded below by (K), any bounded domain (\Omega\subset M), and any CAT(0) space (Y) (no compactness or embedding assumptions), a weak solution (u:\Omega\times(0,\infty)\to Y) with initial data (u_0\in W^{1,2}(\Omega,Y)) and boundary data (\psi\in W^{1,2}(\Omega,Y)) is locally Lipschitz in both space and time. Moreover, the pointwise spatial Lipschitz constant \