Optimal Heinz-type radius bounds for graphs with negative scalar curvature (Chern's 1965 estimate for $\sigma_2\le -c$)

Chern (1965) proved Heinz-type radius bounds for hypersurface graphs in Euclidean space: a graph whose mean curvature satisfies $\sigma_1\ge c>0$ cannot be defined over too large a ball, and — Theorem 4 of that paper — for graphs of dimension $m\ge 3$ with $\sigma_2\le -c<0$, where $\sigma_2$ is the second elementary symmetric function of the principal curvatures (half the scalar curvature), a radius bound $R\le C(m)/\sqrt{c}$ holds, but only under an auxiliary quadratic coercivity condition and with a constant that Chern himself expected to be improvable. The optimal form of this negative-scalar-curvature estimate is still unknown: the problem asks whether the bound holds under $\sigma_2\le -c$ alone, what the sharp constant is when it holds, and whether a complete entire graph in $\mathbb{R}^{m+1}$, $m\ge 3$, can have scalar curvature bounded above by a negative constant — the graph analogue of Efimov's classical theorem, known so far only under auxiliary bounds on the second fundamental form or the mean curvature. The Gauss-map density question raised in the same 1965 paper was answered negatively by Hoffman, Osserman and Schoen (1982), whose hemisphere rigidity theorem then superseded it, so that thread is resolved and the negative-$\sigma_2$ estimate is the sole open core.

Background

Let $u$ be a $C^2$ function on a domain $\Omega\subset\mathbb{R}^m$. Its graph $\Gamma_u=\{(x,u(x)):x\in\Omega\}$ is a hypersurface in $\mathbb{R}^{m+1}$, described at each point by the second fundamental form $h_{ij}$ — the second derivatives of the immersion projected on a unit normal — or equivalently by the shape operator $A$, a self-adjoint endomorphism whose eigenvalues $\kappa_1,\dots,\kappa_m$ are the principal curvatures. The elementary symmetric functions $\sigma_k=\sum_{i_1<\cdots<i_k}\kappa_{i_1}\cdots\kappa_{i_k}$ interpolate between the mean curvature ($\sigma_1$, taken here as the sum of the principal curvatures) and the scalar curvature: by the Gauss equation the scalar curvature of the induced metric equals $2\sigma_2$. The Gauss map sends each point of the graph to its unit normal, a point of the unit sphere $S^m$.

A classical theme, going back to Bernstein's theorem — an entire graph in $\mathbb{R}^3$ whose mean curvature vanishes identically (a minimal graph) is a plane — is that a curvature bound limits the size of the parameter domain of a graph. Heinz (1955) proved that a surface graph over a disk of radius $R$ whose mean curvature is at least $c>0$ satisfies $R\le 1/c$. Chern (1965) reorganized this circle of ideas for hypersurfaces of any dimension through integral identities for a piece of hypersurface whose normals make an angle at most $a$ with a fixed direction, comparing the area $V_a$ of its Gauss image with the area $L_a$ of its projection: when $\sigma_1\ge c$ the identity reads $mc\,V_a\le L_a$, and when $\sigma_2\ge b$ it reads $\sqrt{m(m-1)b}\,V_a\le L_a$. Both imply Heinz-type radius bounds for graphs, and a contemporary remark of Flanders (1966) on the mean-curvature case belongs to the same family, now usually called the Bernstein–Heinz–Chern–Flanders inequalities.

The situation changes when the curvature is bounded above by a negative constant, because the prescribed-curvature equation of the graph degenerates wherever the curvature tends to zero and no elliptic estimate applies. For surfaces, Efimov's classical theorem (Efimov 1968, in English translation) states that no complete immersed surface in $\mathbb{R}^3$ has Gauss curvature $\kappa_1\kappa_2\le -c<0$ everywhere. In higher dimensions the analogue is open: Smyth and Xavier (1987) settled the three-dimensional case of the extension posed independently by Reilly and Yau — a complete hypersurface in $\mathbb{R}^4$ whose Ricci curvature is bounded above by a negative constant satisfies $\inf\|A\|=0$, so that no such hypersurface exists — and obtained only partial results beyond dimension three. For graphs, Chern (1965, Theorem 4) proved a local analogue: a graph over a ball of radius $R$ in $\mathbb{R}^m$, $m\ge 3$, with $\sigma_2\le -c<0$, satisfies a radius bound $R\le C(m)/\sqrt{c}$ — but only under an auxiliary quadratic coercivity condition: a pointwise lower bound on a quadratic expression in the principal curvatures that restores control exactly where the scalar-curvature inequality degenerates. Chern remarked that the constant in this estimate can most likely be improved, and the contemporary zbMATH review of the paper likewise judged the inequality almost certainly non-sharp.

The subsequent literature has approached this estimate only through auxiliary hypotheses. Elbert (2000) proved that no complete graph with bounded $\|A\|$ can have scalar curvature bounded above by a negative constant (the paper treats negative $r$-th mean curvature in general); Hasanis and Vlachos (2004) obtained related rigidity for entire graphs under the same boundedness hypothesis; Fontenele (2010) proved that an entire graph with bounded mean curvature satisfies $\inf|R|=0$ (with $R$ now denoting the scalar curvature), improving results of Elbert and of Hasanis–Vlachos, and that $\inf\|A\|=0$ holds for entire graphs of negative Ricci curvature in dimensions $n\ge 3$, improving a result of Chern and giving a partial answer to a question of Smyth and Xavier; he also records Okayasu's (1989) construction of a complete $O(2)\times O(2)$-invariant hypersurface of constant negative scalar curvature in $\mathbb{R}^4$ — whose unbounded $\|A\|$ shows that the graph hypothesis is essential to the global question — and states explicitly that he does not know whether the bounded-mean-curvature assumption can be removed. Coswosck and Fontenele (2021) generalized and improved these estimates to graphs over complete Riemannian domains, still under Ricci-type hypotheses. On the quantitative side, Fontenele and Silva (2005) determined sharp estimates for the size of balls in the complement of a complete hypersurface with bounded $\|A\|$ — a different, extrinsic setting — and Barbosa, Bessa and Montenegro (2008) extended the Bernstein–Heinz–Chern–Flanders inequalities to foliations. No known work removes Chern's coercivity condition for $m\ge 3$, determines the sharp constant of the $\sigma_2\le -c$ radius bound, or settles the existence of complete entire graphs with uniformly negative scalar curvature without auxiliary hypotheses.

The Gauss-map question raised in the 1965 paper is resolved. Chern (1965) raises there, on p. 83, the density question — whether the Gauss map of a complete surface with non-zero constant mean curvature is everywhere dense. Hoffman, Osserman and Schoen (1982) answered it negatively: the unduloids (complete constant-mean-curvature surfaces of revolution) have Gauss images in an arbitrarily narrow strip about a great circle, and such a strip omits open sets. The plane/cylinder rigidity is Theorem 1 of Hoffman–Osserman–Schoen (1982), under the hypothesis that the Gauss image lies in a closed hemisphere, and it was later generalized to uniformly elliptic Weingarten surfaces — surfaces whose principal curvatures satisfy a fixed elliptic relation — by Fernández, Gálvez and Mira (2023). This Gauss-map thread is resolved and contributes no open core; the open core is the negative-$\sigma_2$ radius estimate described above.

Problem Statement

Determine the optimal Heinz-type radius estimate for hypersurface graphs with uniformly negative scalar curvature — the open core of Chern's 1965 Theorem 4. Throughout, $\sigma_2=\sum_{i<j}\kappa_i\kappa_j$ denotes the second elementary symmetric function of the principal curvatures of a graph in $\mathbb{R}^{m+1}$ with respect to the upward unit normal (equal to half the scalar curvature of the induced metric), and $m\ge 3$. For $m=2$, where $\sigma_2$ is the Gauss curvature, the coercivity-free bound is classical — Heinz's estimate $\inf|\sigma_2|\le 3e^2/R^2$ (with $e$ Euler's number) for a graph over a disk of radius $R$ yields $R\le e\sqrt{3/c}$ — which is why the problem below concerns $m\ge 3$.

(1) Coercivity-free local bound. Prove or disprove: there exists a constant $C(m)$, depending only on $m$, such that for every $c>0$ and every $C^2$ function $u$ on the closed ball $\overline{B}_R(0)\subset\mathbb{R}^m$ whose graph has $\sigma_2\le -c$ at every point, one has $R\le C(m)/\sqrt{c}$. Chern proved this only under an additional quadratic coercivity condition on the principal curvatures; the statement above imposes no coercivity, boundedness, or curvature-growth condition.

(2) Sharp constant. To the extent the bound in (1) holds, determine the optimal value of $C(m)$: prove the bound for that value and exhibit a family of graphs over balls of radii tending to $C(m)/\sqrt{c}$ from below, all with $\sigma_2\le -c$, showing that no smaller constant is possible. An explicit constant strictly smaller than Chern's, even under a weakened coercivity condition, does not determine the optimal value of $C(m)$.

(3) Global counterpart. Settle whether there exists a complete entire graph in $\mathbb{R}^{m+1}$, $m\ge 3$ — a $C^2$ function on all of $\mathbb{R}^m$ whose graph, with its induced metric, is complete — with $\sigma_2\le -c<0$ everywhere for some $c>0$.

These three components share one scientific objective: the optimal local-to-global control of the parameter domain of a hypersurface graph under a uniformly negative scalar-curvature bound. They determine one another: a proof in (1) implies nonexistence in (3), because a complete entire graph restricts to graphs over balls of every radius; conversely, an affirmative construction in (3), or a sequence of graphs over balls of arbitrarily large radius with $\sigma_2\le -c$ for fixed $c$, refutes (1) and shows that no finite constant exists in (2).

References

1. S.-S. Chern, On the curvatures of a piece of hypersurface in euclidean space, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 29 (1965), 77–91. DOI: 10.1007/BF02996311, https://doi.org/10.1007/BF02996311
2. E. Heinz, Über Flächen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind, Mathematische Annalen 129 (1955), 451–454. DOI: 10.1007/BF01362385, https://doi.org/10.1007/BF01362385
3. H. Flanders, Remark on Mean Curvature, Journal of the London Mathematical Society s1-41 (1966), 364–366. DOI: 10.1112/jlms/s1-41.1.364, https://doi.org/10.1112/jlms/s1-41.1.364
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13. F. A. Coswosck, F. Fontenele, Curvature estimates for graphs over Riemannian domains, The Journal of Geometric Analysis 31 (2021), 5687–5720. DOI: 10.1007/s12220-020-00497-w, https://doi.org/10.1007/s12220-020-00497-w
14. I. Fernández, J. A. Gálvez, P. Mira, Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs, American Journal of Mathematics 145 (2023), 1887–1921. DOI: 10.1353/ajm.2023.a913297, https://doi.org/10.1353/ajm.2023.a913297; preprint arXiv:2004.08275, https://arxiv.org/abs/2004.08275
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