The Chern-Goldberg volume-decreasing problem for real harmonic maps: the sign of the invariant $C$ and the removal of the hypothesis $C \le 0$

Chern and Goldberg (1975) studied equidimensional harmonic maps $f: M \to N$ between oriented Riemannian $n$-manifolds and proved that such a map is volume decreasing -- the squared Jacobian $u = (\det df)^2$ satisfies $u \le 1$ -- when the domain is a hyperbolic ball of sectional curvature $-4$ or Euclidean space and the target is an Einstein manifold of sufficiently negative scalar curvature, provided a scalar invariant $C$, built quadratically from the second covariant derivatives of $f$ contracted with the adjugate of $df$, satisfies $C \le 0$; in the Euclidean case the same hypotheses force $u \equiv 0$. The paper computes $C$ in only two cases ($C = 0$ for planar harmonic maps given by conjugate harmonic functions; $C \ge 0$ for Gauss maps of constant-mean-curvature surfaces in $\mathbb{R}^3$) and leaves open the sign of $C$ in general and whether $C \le 0$ can be removed or replaced by a checkable geometric condition. No work to date determines the sign of $C$ in general or proves the volume-decreasing theorems under curvature bounds alone; the problem remains open.

Background

A smooth map $f: M \to N$ between Riemannian manifolds is harmonic if it is a critical point of the Dirichlet energy $\frac{1}{2}\int_M |df|^2$, equivalently if its tension field $\tau(f) = \operatorname{tr}\nabla df$ vanishes identically. Harmonic maps generalize geodesics and holomorphic maps, and the classical Schwarz lemma for holomorphic self-maps of the unit disk has long served as the model for rigidity statements about mappings between curved spaces. For equidimensional holomorphic maps between Hermitian manifolds, Chern (1968) initiated the volume-decreasing theory (with a formula later corrected in the Chern-Goldberg paper), and Yau (1978) proved the general Schwarz lemma for Kähler manifolds (complex manifolds whose complex structure is compatible with the Riemannian metric) that became the standard result of that theory. The 1975 Chern-Goldberg paper asks how far the analogous statements survive for real -- that is, not necessarily holomorphic -- harmonic maps between Riemannian manifolds of the same dimension.

The setup is as follows. Let $f: M \to N$ be a harmonic map between oriented Riemannian $n$-manifolds. In local orthonormal coframes $\{\omega^i\}$ on $M$ and $\{\omega^\alpha\}$ on $N$ one writes $f^*\omega^\alpha = \sum_i a_{\alpha i}\omega^i$, so the matrix $a = (a_{\alpha i})$ represents the differential $df$ and $u = (\det a)^2$ is the squared volume ratio of $f$ -- the natural quantity for a volume-decreasing statement. Let $B = (B_{i\alpha})$ be the adjugate of $a$ (the transpose of the cofactor matrix, so that $aB = (\det a)I$), let $a_{\alpha i j}$ denote covariant derivatives of the coefficients, and let $A_k = \nabla_k \det a = \sum_{\alpha,i} B_{i\alpha} a_{\alpha i k}$ denote the covariant derivative of the Jacobian $\det a$ (so that $\sum_k A_k^2 = |\nabla \det a|^2$). The paper's fundamental Bochner-type identity (a formula for the Laplacian of $u$ obtained by commuting covariant derivatives), in the form corroborated by later re-derivations, is

$\frac{1}{4}\Delta u = \sum_k A_k^2 + \frac{u}{2}\Big(R - \sum_{\beta,\gamma,k} a_{\beta k}a_{\gamma k}R^*_{\beta\gamma}\Big) - \frac{C}{2},\qquad C = \sum_{i,\alpha,l,\beta,j} B_{i\alpha}B_{l\beta}\,a_{\alpha l j}a_{\beta i j},$

with the logarithmic companion $\frac{1}{2}\Delta\log u = R - \sum_{\beta,\gamma,k}a_{\beta k}a_{\gamma k}R^*_{\beta\gamma} - C/u$, where $R$ and $R^*$ are the scalar curvatures of $M$ and $N$ and $R^*_{\beta\gamma}$ is the Ricci tensor of $N$. The first term on the right is nonnegative; on a negatively curved domain the curvature combination can be made nonnegative by taking the target Ricci curvature negative enough. The last term has a definite sign only under the hypothesis $C \le 0$, and this is exactly where the maximum-principle argument needs it. Under $C \le 0$ the paper proves, in the normalizations of its statements: a harmonic map from a ball in the simply connected space form of sectional curvature $-4$ (a space form is a complete Riemannian manifold of constant sectional curvature, so this is hyperbolic $n$-space rescaled to curvature $-4$) into an Einstein manifold (a Riemannian manifold whose Ricci tensor is proportional to the metric) with scalar curvature $\le -4n(n-1)$ satisfies $u \le 1$; a harmonic map from flat Euclidean $n$-space into an Einstein manifold with scalar curvature bounded above by a negative constant satisfies $u \le 1$; and in the Euclidean case the map is in fact totally degenerate, $u \equiv 0$. These are sharp analogues of the Schwarz lemma at the volume level, but none is unconditional: $C$ is a quadratic expression in the second derivatives of the map contracted with the adjugate of its first derivatives, and no curvature hypothesis known at the time controls its sign. The paper computes $C$ in only two cases: $C = 0$ for harmonic self-maps of the Euclidean plane given by pairs of conjugate harmonic functions, and $C \ge 0$ for Gauss maps of surfaces of constant mean curvature in $\mathbb{R}^3$ -- the wrong sign for the volume-decreasing application, though favorable for superharmonicity: by the Ruh-Vilms theorem the Gauss map of a constant-mean-curvature surface (the map sending each point to its unit normal, viewed as a point of the unit sphere; constant mean curvature means the average of the two principal curvatures is constant) is harmonic, and for it $u = K^2$ with $K$ the Gauss curvature, so the identity makes $\log K^2$ superharmonic (nonpositive Laplacian) wherever $K \ne 0$. The paper derives its own sphere-type characterization in this setting but states that no further geometrical conclusion could be drawn from that superharmonicity.

Two neighboring bodies of work frame the problem without settling it. Geometrically, Klotz and Osserman (1966) proved that a complete surface of constant mean curvature in $\mathbb{R}^3$ whose Gauss curvature keeps one sign is a sphere or a right circular cylinder, and Hoffman, Osserman and Schoen (1982) proved that such a surface whose Gauss map image lies in an open hemisphere is a plane (with image in a closed hemisphere: a plane or a right cylinder), with unduloids (the noncompact constant-mean-curvature surfaces of revolution in $\mathbb{R}^3$) showing sharpness -- using the harmonicity of the Gauss map but not the invariant $C$. Analytically, Hildebrandt, Jost and Widman (1980) obtained Schwarz-type Lipschitz bounds for harmonic maps whose image lies in a sufficiently small geodesic ball; and a line of work from Goldberg-Ishihara (1976) through Har'El (1978), Goldberg-Har'El (1979) and Shen (1984) proved distance-decreasing Schwarz lemmas for harmonic maps of bounded dilatation (the conformal distortion of $df$, that is, the ratio of its largest to its smallest singular value). This dilatation line was extended to V-harmonic maps -- a generalization of harmonic maps, introduced by Chen-Jost-Qiu (2012), in which an external vector field enters the Euler-Lagrange equation -- by Chen-Zhao (2017), Chen-Li-Qiu (2022) and Zhao (2026), while Zhao (2018) proved the volume-decreasing theorem for V-harmonic maps that is the direct descendant of the Chern-Goldberg theorem and still imposes on its hypotheses a contraction of second derivatives with the adjugate of the differential that is algebraically identical to $C \le 0$. The dilatation results replace the uncontrolled second-order condition by a first-order distortion hypothesis; none determines the sign of $C$ or removes the condition, and the Jacobian question under curvature bounds alone remains untouched.

Problem Statement

Let $f: M \to N$ be a harmonic map (a critical point of the Dirichlet energy $\frac{1}{2}\int_M |df|^2$, equivalently a map whose tension field $\operatorname{tr}\nabla df$ vanishes identically) between oriented Riemannian $n$-manifolds of the same dimension. In local orthonormal coframes (bases of 1-forms dual to orthonormal tangent frames) write $f^*\omega^\alpha = \sum_i a_{\alpha i}\omega^i$, let $a = (a_{\alpha i})$ be the matrix of the differential, $u = (\det a)^2$ the squared Jacobian (volume ratio), $B = (B_{i\alpha})$ the adjugate of $a$, $a_{\alpha i j}$ the covariant derivatives of the coefficients, and

$C = \sum_{i,\alpha,l,\beta,j} B_{i\alpha}B_{l\beta}\,a_{\alpha l j}\,a_{\beta i j}.$

The Chern-Goldberg volume-decreasing theorems state, under the hypothesis $C \le 0$, that $u \le 1$ for harmonic maps from a ball in the simply connected space form of sectional curvature $-4$ (hyperbolic $n$-space rescaled to constant sectional curvature $-4$) into an Einstein manifold (a manifold whose Ricci tensor is proportional to its metric) of scalar curvature $\le -4n(n-1)$, and that $u \le 1$ -- in fact $u \equiv 0$, total degeneracy -- for harmonic maps from flat Euclidean $n$-space into an Einstein manifold of scalar curvature bounded above by a negative constant. The problem is to complete this program:

(1) Sign of $C$. Consider only nondegenerate maps -- those with $\nabla df \not\equiv 0$ and $\operatorname{rank} df = n$ at some point, so that $C$ does not vanish for structural reasons (total geodesy or a rank-deficient adjugate). Determine the sign of $C$ for natural classes of such harmonic maps. Find geometrically checkable conditions -- on the curvatures of $M$ and $N$, on the map, or on the second-order behavior of $f$ -- that force $C \le 0$; or show that $C$ can be positive in the settings of the theorems above. The two cases computed in the source ($C = 0$ for planar harmonic maps given by pairs of conjugate harmonic functions, the real and imaginary parts of a holomorphic function; $C \ge 0$ for Gauss maps of constant-mean-curvature surfaces) remain the extent of the known computations.

(2) Removal or relaxation of the hypothesis (in dimensions $n \ge 3$). Decide whether the volume-decreasing conclusions hold under curvature bounds alone, that is, without any hypothesis on $C$: does a harmonic map from the hyperbolic ball of curvature $-4$ (respectively from Euclidean $n$-space) into an Einstein manifold with scalar curvature $\le -4n(n-1)$ (respectively $\le -\varepsilon^2 < 0$) necessarily satisfy $u \le 1$ (respectively $u \equiv 0$)? A positive answer may control $C$ through the equations themselves or use an entirely different argument; a negative answer is an explicit equidimensional harmonic map satisfying the curvature hypotheses with $u > 1$ somewhere (refuting the hyperbolic-ball statement) or with $u > 0$ somewhere (refuting the Euclidean statement $u \equiv 0$). Alternatively, formulate and prove a replacement hypothesis that is strictly weaker than $C \le 0$ -- strictly weaker meaning that every equidimensional harmonic map with $C \le 0$ satisfies the new hypothesis but not conversely -- checkable, meaning expressible solely in terms of the curvatures of $M$ and $N$ and first-order data of $f$ other than the volume ratio $u = (\det a)^2$ itself (equivalently the Jacobian $\det df$), and such that the volume-decreasing conclusions hold under it ($u \le 1$ for the hyperbolic-ball statement, $u \equiv 0$ for the Euclidean statement).

(3) Gauss-map instance. For a surface of constant mean curvature in $\mathbb{R}^3$ (constant mean curvature means the average of the two principal curvatures is constant), the Gauss map (sending each point of the surface to its unit normal, a point of the unit sphere) is harmonic with $u = K^2$ and $C \ge 0$, so $\log K^2$ is superharmonic -- its Laplacian is nonpositive -- wherever $K \ne 0$. Draw from this superharmonicity a new geometric conclusion about complete constant-mean-curvature surfaces that is quantitative -- an explicit estimate or inequality for such a surface, for example a bound on the possible growth of its Gauss curvature -- rather than one of the qualitative classifications of Klotz-Osserman or Hoffman-Osserman-Schoen, which were obtained by other methods.

References

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