Yau's open questions for positive-curvature surfaces in $\mathbb{R}^3$: the $n=2$ case of the pinched-curvature diameter bounds (Theorem 8) and boundedness of surfaces with constant positive Gaussian curvature

In 'Some global theorems on non-complete surfaces' (Comment. Math. Helv. 48 (1973) 177-187), Shing-Tung Yau proved global theorems for surfaces without completeness and explicitly left two questions open for surfaces ($n=2$) in $\mathbb{R}^3$. First, Theorem 8 of that paper bounds the outer diameter ($\le 2\sqrt{K_2}/K_1$) and inner diameter ($\ge 2\sqrt{K_1}/K_2$) of a compact hypersurface in $\mathbb{R}^{n+1}$, $n\ge 3$, whose sectional curvatures lie in $[K_1,K_2]\subset(0,\infty)$, but its proof uses three distinct principal directions and Yau writes: 'We do not know whether Theorem 8 is true for $n=2$ or not.' Second, after observing that the universal cover of a twice-punctured sphere (constant positive curvature, intrinsically unbounded) shows the finiteness hypothesis of his Theorem 7 is essential, he writes: 'it is quite possible that any surface with constant positive curvature in euclidean three-space is bounded.' Adjacent results exist under stronger hypotheses (Pogorelov's nearly spherical surfaces; Gálvez–Martínez's estimates for constant-curvature surfaces with planar boundary; Gálvez–Hauswirth–Mira's classification of isolated singularities; Smith's classification of quasicomplete $k$-surfaces (arXiv 2022, published 2024)), but neither question has been resolved in the literature. Both questions remain open.

Background

An isometric immersion of a Riemannian surface $M$ into $\mathbb{R}^3$ is a smooth map that preserves the lengths of curves; one says $M$ is 'a surface in euclidean three-space'. The Gaussian curvature $K$ of the immersed surface is the product $\lambda_1\lambda_2$ of the two principal curvatures, i.e. of the eigenvalues $\lambda_1,\lambda_2$ of the shape operator (the derivative, along the surface, of the map sending each point of the surface to its unit normal vector in $\mathbb{R}^3$). For a hypersurface of $\mathbb{R}^{n+1}$ with $n\ge 3$, the sectional curvature of the 2-plane spanned by two principal directions (eigendirections of the shape operator) equals $\lambda_i\lambda_j$ (Gauss equation), so a pinching condition $K_1\le\lambda_i\lambda_j\le K_2$ on all sectional curvatures is weaker than a pinching of the principal curvatures themselves. A surface has strongly positive curvature if $K$ is bounded below by a positive constant. An end of a finitely connected surface is a puncture of a compact surface it is diffeomorphic to; a surface is parabolic if it admits no non-constant negative subharmonic function; the total curvature is $\int_M K\,dA$, and the classical Cohn-Vossen inequality for complete surfaces states $\int K\,dA\le 2\pi\chi$ ($\chi$ = Euler characteristic). Bonnet's theorem (in its Myers form) states that a complete surface with $K\ge\delta>0$ is compact with diameter $\le\pi/\sqrt{\delta}$; Liebmann's classical theorem states that the only compact surface in $\mathbb{R}^3$ with constant positive Gaussian curvature is the round sphere. For a compact surface $M\subset\mathbb{R}^3$, following Yau, the outer diameter of $M$ is 'the smallest diameter of balls containing $M$' and the inner diameter is 'the largest diameter of balls lying inside $M$' (inside the body $M$ bounds; a compact surface with $K>0$ has definite second fundamental form, hence is locally strictly convex — every point has a neighborhood that bounds a convex cap — and the classical Hadamard-type theory of locally convex surfaces then makes $M$ a convex embedded sphere). Yau's 1973 paper proves, among other results: (Theorem 2 and its corollary) a non-positively curved, finite-volume, parabolic surface in $\mathbb{R}^3$ with at most three ends lies in a plane, hence a complete non-positively curved surface of finite volume with at most three ends cannot be immersed in $\mathbb{R}^3$; (Theorems 3-6) Cohn-Vossen/Huber-type inequalities $\int K\,dA\ge 2\pi\chi$ for parabolic surfaces with finiteness conditions at the ends; (Theorem 7) a surface with strongly positive curvature admitting, near each end, closed curves of uniformly bounded length is bounded (finite intrinsic diameter — the proof is a Bonnet argument on minimizing geodesics); and (Theorem 8) for compact hypersurfaces in $\mathbb{R}^{n+1}$, $n\ge 3$, with sectional curvatures in $[K_1,K_2]$, $0<K_1\le K_2$, the outer diameter is at most $2\sqrt{K_2}/K_1$ and the inner diameter at least $2\sqrt{K_1}/K_2$ (both equal to $2/\sqrt{K}$, sharp, for the round sphere of curvature $K_1=K_2=K$). The $n\ge 3$ proof derives from $K_1\le\lambda_i\lambda_j\le K_2$ the pointwise principal-curvature bound $\lambda_i\ge K_1/\sqrt{K_2}$ — using three distinct principal directions — and then applies a plane-curve theorem of Yu. S. Bogdanov: a closed convex plane curve whose curvature is everywhere at least $k$ has outer diameter at most $2/k$; for $n=2$ the first step is unavailable, since Gaussian-curvature pinching does not control the ratio $\lambda_1/\lambda_2$ pointwise. Two further notions from the later literature: peaked spheres of Gálvez–Hauswirth–Mira are constant-positive-curvature surfaces in $\mathbb{R}^3$ with prescribed isolated conical singularities; and Smith's quasicompleteness is completeness of the metric $I+III$, where $I$ is the first fundamental form (the metric induced by the immersion) and $III$ is the third fundamental form (the metric $III(X,Y)=\langle dN(X),dN(Y)\rangle$ induced by the Gauss map $N$); this is weaker than completeness of $I$ but still a genuine hypothesis. Classical context: by Hilbert's and Efimov's theorems, complete surfaces in $\mathbb{R}^3$ with sufficiently negative curvature do not exist, and Yau's Theorem 2 plays the analogous non-immersibility role for non-positive curvature with few ends and finite volume; the questions concern the positive-curvature, non-complete side of the same circle of ideas.

Problem Statement

Both parts concern smooth ($C^2$ or smoother) isometrically immersed surfaces in euclidean three-space $\mathbb{R}^3$. Part A (the $n=2$ case of Yau's Theorem 8): Let $M$ be a compact (closed, without boundary) surface immersed in $\mathbb{R}^3$ whose Gaussian curvature satisfies $K_1\le K\le K_2$ everywhere, with $0<K_1\le K_2$. (Such a surface is, by classical Hadamard-type theory of locally convex surfaces, a convex embedded sphere, so 'balls lying inside $M$' means balls in the convex body it bounds.) (i) Is the outer diameter of $M$ — the diameter of the smallest euclidean ball containing the image of $M$ — at most $2\sqrt{K_2}/K_1$? (ii) Is the inner diameter of $M$ — the diameter of the largest euclidean ball contained in the body bounded by $M$ — at least $2\sqrt{K_1}/K_2$? Both constants are sharp for the round sphere ($K_1=K_2=K$: outer and inner diameter both $2/\sqrt{K}$), and the known trivial bound from Bonnet–Myers is outer diameter $\le\pi/\sqrt{K_1}$, so the content of (i) is the improvement under the upper curvature bound $K_2$. Part B (Yau's boundedness conjecture): Does there exist a smooth isometric immersion into $\mathbb{R}^3$ of a connected surface whose Gaussian curvature is identically a positive constant, such that the image of the immersion is unbounded in $\mathbb{R}^3$? If not, then every surface of constant positive Gaussian curvature in $\mathbb{R}^3$ has bounded image, as Yau conjectured ('it is quite possible that any surface with constant positive curvature in euclidean three-space is bounded'). Here 'bounded' is read extrinsically — the image is contained in some euclidean ball — the only reading consistent with Yau's own example, since the universal cover of a twice-punctured round sphere, with its induced constant-positive-curvature metric, is isometrically immersed in $\mathbb{R}^3$ via the covering map and is intrinsically unbounded while its image is bounded. Part B must be answered with no completeness, quasicompleteness, properness, embeddedness, or end-finiteness hypothesis beyond constancy of the (positive) Gaussian curvature; in particular any proof must be compatible with the existence of the bounded covering immersion of the universal cover of a twice-punctured round sphere, and any counterexample must have constant positive curvature and unbounded image.

References

1. Yau, Shing-Tung. 'Some global theorems on non-complete surfaces.' Commentarii Mathematici Helvetici 48 (1973): 177-187. DOI 10.1007/BF02566120. https://doi.org/10.1007/BF02566120 (Primary source.)
2. Pogorelov, A. V. 'Nearly spherical surfaces.' Journal d'Analyse Mathématique 19 (1967): 313-321. DOI 10.1007/BF02788724. https://doi.org/10.1007/BF02788724 (Content as described in Yau's Remark 2: a diameter-type theorem for surfaces whose ratio of principal curvatures is close to 1.)
3. Guggenheimer, H. 'Nearly spherical surfaces.' Aequationes Mathematicae 3 (1969): 186-193. DOI 10.1007/BF01817512. https://doi.org/10.1007/BF01817512
4. Gulliver II, R. D.; Osserman, R.; Royden, H. L. 'A theory of branched immersions of surfaces.' American Journal of Mathematics 95 (1973): 750-812. DOI 10.2307/2373697. https://doi.org/10.2307/2373697 (Yau's acknowledgment states this paper also discusses Ogawa's four-ended example.)
5. Gálvez, José A.; Hauswirth, Laurent; Mira, Pablo. 'Surfaces of constant curvature in R^3 with isolated singularities.' Advances in Mathematics 241 (2013): 103-126. DOI 10.1016/j.aim.2012.11.019. arXiv:1007.2523. https://arxiv.org/abs/1007.2523 (Classification of isolated/conical singularities, including peaked spheres. Title given in the arXiv form; the published title renders R^3 in double-struck type.)
6. Gálvez, José A.; Martínez, Antonio. 'Estimates in surfaces with positive constant Gauss curvature.' Proceedings of the American Mathematical Society 128 (2000): 3655-3660. DOI 10.1090/S0002-9939-00-05805-6. https://doi.org/10.1090/S0002-9939-00-05805-6 (Optimal height, area and volume bounds for compact constant-curvature surfaces with planar boundary, spherical caps as extremals.)
7. Gálvez, José A.; Martínez, Antonio; Teruel, José L. 'Complete surfaces with ends of non positive curvature.' arXiv:1405.0851. https://arxiv.org/abs/1405.0851 (Extension of Efimov's theorem to surfaces whose ends have non-positive curvature; work on Milnor's conjecture; does not concern the positive-curvature questions here.)
8. Smith, Graham. 'On quasicomplete $k$-surfaces in 3-dimensional space-forms.' Bulletin of the London Mathematical Society 56 (2024): 1351-1361. DOI 10.1112/blms.12998. arXiv:2211.14868. https://arxiv.org/abs/2211.14868 (Defines quasicompleteness as completeness of $I+III$; proves that the only quasicomplete immersed surfaces of constant extrinsic curvature $k>\max(0,-c)$ in the space-form of curvature $c$ are geodesic spheres.)
9. Bogdanov, Yu. S. 'Some tests for the absence of closed trajectories' [in Russian; English title per zbMATH]. Doklady Akademii Nauk SSSR 120, no. 5 (1958): 939-940. MR 0098871. https://www.mathnet.ru/eng/dan23140 (This is reference [4] of Yau (1973), where the author's name is misprinted as 'Bogdana'; source of the closed-convex-plane-curve outer-diameter bound — curvature at least $k$ implies outer diameter at most $2/k$ — used in the proof of Theorem 8.)