Yau's Problem 77 in Differential Geometry

• Yau's Problem 77 is a set of deep conjectures in Riemannian and complex differential geometry focused on rigidity and classification issues.
• It encompasses topics like uniformization, sphere theorems, spectral analysis, and CR fillability, employing tools such as holomorphic flows and Ricci flow.
• Recent advances use techniques from eigenvalue analysis and curvature pinching to partially resolve conjectures and reveal new analytic obstructions.

Yau's Problem 77

Yau's Problem 77 refers to a prominent collection of deep and diverse open questions, conjectures, and rigidity theorems across Riemannian and complex differential geometry, CR geometry, geometric analysis, and global analysis. The designation "Problem 77" is context-dependent; it refers to problem numbering in Yau’s lists of open problems (notably those from the 1980s and 1990s) and encompasses several outstanding geometric rigidity and classification conjectures—most famously uniformization, eigenvalue, pinching, Li–Yau inequalities, and Plateau-type problems.

1. Uniformization and Complex Analytic Rigidity

The original "Yau's Uniformization Conjecture" posits that any complete, noncompact Kähler manifold $(M^n,g)$ with strictly positive holomorphic bisectional curvature is biholomorphic to $\mathbb{C}^n$. In (Liu, 2016), Liu confirms this conjecture under the additional hypothesis of maximal volume growth. Specifically, if $M^n$ has nonnegative holomorphic bisectional curvature and for some $v>0$ and all $r > 0$,

$\operatorname{Vol} B(p, r) \geq v r^{2n}$

for some (hence any) $p \in M$, then $M$ is biholomorphic to $\mathbb{C}^n$.

Key technical tools:

• Construction of a global holomorphic vector field $X$ whose real part produces a contracting flow to a unique fixed point.
• Application of Cheeger–Colding Gromov–Hausdorff convergence: the rescaled pointed limit as $r_i \to \infty$ is a metric cone. Hörmander $L^2$-methods on Nakano-positive bundles permit the correction of approximately holomorphic gradient-like vector fields to genuine holomorphic ones.
• Construction of $n$ algebraically independent holomorphic functions of polynomial growth, defining a biholomorphism to $\mathbb{C}^n$.

The general case without the maximal volume growth assumption remains open, with obstacles including non-collapsing issues and analytic singularities of tangent cones (Liu, 2016).

2. Pinching Problems and Sphere Theorems

Yau’s Problem 77 in pinching theory, as formulated in (Li, 29 Aug 2025), centers on whether sharp sphere theorems can be proven when curvature pinching is formulated in terms of scalar curvature rather than sectional maxima. Let $(M^n, g)$ be closed and simply connected, and let $K_\text{min}$, $K_\text{max}$ denote minimum and maximum sectional curvatures, respectively. The classical result is that $K_{\min}/K_{\max} > 1/4$ implies $M \simeq S^n$ topologically; Yau asked whether the normalized scalar curvature $S_0 = \frac{S}{n(n-1)}$ can replace $K_{\max}$.

Key developments:

• Li (Li, 29 Aug 2025) proves a topological sphere theorem under the four-frame algebraic curvature pinching

$$R_{1313} + R_{1414} + R_{2323} + R_{2424} > \frac{1}{2}(R_{1212} + R_{3434}),$$

strictly weaker than $1/4$-pinching. He further demonstrates that the scalar-curvature-based pinching

$K_{\min} > \frac{n(n-1)}{n^2-n+12} S_0$

suffices for homeomorphism to $S^n$. In dimensions $n=4$ and $n\geq 12$, diffeomorphism follows by Ricci flow, while the case $5 \leq n \leq 11$ remains open.

Technical ingredients: algebraic curvature reductions, positive isotropic curvature (PIC) and its implications for $\pi_i(M)=0$, Ricci flow preservation of curvature cones, and de Rham-type splitting arguments.

Sharpness: Constants are not optimal; the boundary value $(n-1)/(n+2)$ is realized by projective space.

3. Spectral Geometry and First Eigenvalue on Minimal Hypersurfaces

Yau conjectured that for every closed embedded minimal hypersurface $M^n$ in the sphere $S^{n+1}(1)$, the first positive Laplacian eigenvalue $\lambda_1$ equals $n$. Tang and Yan, and in particular (Tang et al., 2012), confirm the conjecture in substantial new cases:

• For all minimal homogeneous isoparametric hypersurfaces with $g=6$, $\lambda_1 = n$.
• For certain focal submanifolds of the OT–FKM (Clifford) family with $g=4$ and for $g=6$, the first eigenvalue is proved to be the dimension.
• In the remaining cases, precise lower and upper bounds are established.

Methodology combines min–max characterization of eigenvalues, variational methods with Dirichlet domains, volume distortion formulas, integration over normal bundles, and explicit construction of test functions using Clifford systems.

4. Nodal Sets, Li–Yau Inequalities, and the Heat Equation

In the spectral context, Yau proposed (Problem 77 in the context of nodal sets) lower and upper estimates for the $(n-1)$-dimensional Hausdorff measure of nodal sets of Laplace eigenfunctions on a compact Riemannian manifold. Logunov (Logunov, 2016) established the lower bound:

$$c \sqrt{\lambda} \leq \mathcal{H}^{n-1}(\{\varphi_\lambda = 0\}),$$

where $c = c(M, g)$ and $\lambda$ is the eigenvalue. Key technical innovations include combinatorial partitioning of local charts, doubling index arguments, and refined control over sign changes—obviating reliance on analyticity (Logunov, 2016).

For heat equation gradient estimates, Xia (Xia, 2023) solves Problem 77 for Finsler (possibly non-reversible) geometries, providing generalized Li–Yau inequalities and their sharp Harnack inequalities under $\operatorname{Ric}_N \geq K$. Moreover, she establishes equivalence between the curvature-dimension bound and Li–Yau-type gradient inequalities in the linearized Finsler heat flow.

5. The Complex Plateau Problem and CR Fillability

Another central component of Yau's Problem 77 addresses smoothability criteria for Stein fillings of boundaries of strictly pseudoconvex CR manifolds. Yau [1981] established that for real dimension $2n-1\geq 5$ $(n\geq3)$, vanishing of the Kohn–Rossi cohomology $H^{p,q}_{KR}(X) = 0$ for $1\leq q\leq n-2$ sufficed for the boundary to bound a smooth complex hypersurface (Du, 2017).

Du–Yau and subsequent work by Du–Gao–Yau (Fernex, 2018) distilled this to a single numerical invariant $g^{(1)}(M)$: the cokernel of the wedge map from global CR-holomorphic 1-forms to $n$-forms. The vanishing of $g^{(1)}(M)$ is necessary and sufficient for smooth fillability (up to normalization).

Further, in resolution theory of Stein varieties, link-theoretic invariants detect whether an isolated normal singularity is smooth, with the link's CR structure containing complete information (Fernex, 2018).

6. Calabi–Yau Geometry, Hull–Strominger System, and Futaki Invariants

Yau's Problem 77 in the context of the Hull–Strominger system (heterotic string theory) concerns the existence of special balanced Hermitian metrics satisfying an anomaly cancellation condition. Classical necessary conditions are balancedness, Chern class matching, and slope-polystability for bundles (Garcia-Fernandez et al., 2023). Garcia-Fernandez and González Molina (Garcia-Fernandez et al., 2023) show these are not sufficient, constructing a family of Bott–Chern Futaki invariants on the solution algebroid bundle whose vanishing is an additional necessary condition. This provides the first analytic obstruction beyond the classical topological and stability constraints.

7. Further Directions and Known Failures

Counterexamples:

Yang (Yang, 2011) constructs explicit counterexamples to Yau's higher-dimensional Cohn–Vossen conjecture for Ricci curvature: for $n>2$, in the class of $U(n)$-invariant complete nonnegatively curved Kähler metrics on $\mathbb{C}^n$, the suitably normalized integrals of higher $\sigma_k$ of Ricci can diverge, even for Euclidean volume growth metrics. Under certain decay, the scalar curvature total integral remains bounded, which aligns with the classical Cohn–Vossen for $k=1$.

Asymptotically Conical Kähler Orbifolds:

Faulk (Faulk, 2018) resolves existence and uniqueness of Ricci-flat (Calabi–Yau) metrics on crepant resolutions of Calabi–Yau cones with prescribed decay at infinity, solving a "Yau-type" existence problem via weighted Hölder theory and sharp control of the exceptional spectrum of the Laplacian.

Minimal Hypersurfaces:

For closed Riemannian manifolds with positive Ricci curvature, Marques and Neves (Marques et al., 2013) prove the existence of infinitely many embedded closed minimal hypersurfaces, confirming Yau's conjecture in dimensions up to seven under the Frankel intersection property.

Summary Table: Representative Resolutions of Yau's Problem 77

Geometric Setting	Problem 77 Formulation	Main Results and Techniques	Reference
Noncompact Kähler, $\mathrm{BK} \geq 0$	Uniformization to $\mathbb{C}^n$	Biholomorphism under max vol	(Liu, 2016)
Compact manifolds, Pinching conditions	Sphere theorems via scalar/sectional pinching	Weak 4-frame inequality $\to$ PIC	(Li, 29 Aug 2025)
Eigenvalues of minimal hypersurfaces	$\lambda_1(M) = \dim M$ in $S^{n+1}(1)$	Classification, test functions	(Tang et al., 2012)
Nodal sets, Laplace eigenfunctions	Lower bounds on nodal $(n-1)$-measure	Combinatorial/doubling method	(Logunov, 2016)
Heat equation on Finsler spaces	Li–Yau inequalities and Harnack estimates	Linearized semigroup methods	(Xia, 2023)
Complex Plateau Problem, CR geometry	Cohomological CR invariants for fillability	$g^{(1)}(M)$, normalization	(Fernex, 2018)
Hull–Strominger system, balanced CY	Existence and extended obstructions	Futaki-type invariants	(Garcia-Fernandez et al., 2023)
Ricci curvature and Cohn–Vossen type	Total normalized curvature integrals	Counterexamples for $k\geq2$	(Yang, 2011)
Minimal hypersurfaces, Ricci $>0$	Infinite closed minimal hypersurfaces	Min–max/Almgren–Pitts theory	(Marques et al., 2013)

Conclusion

Yau’s Problem 77 encompasses a broad spectrum of rigidity, existence, and classification problems spanning real, complex, and metric geometry. Substantial advances have been made by refining curvature and spectral pinching techniques, exploiting the geometry of holomorphic and CR structures, and developing new invariants to detect the existence or obstruction of geometric structures. Yet, the boundaries of the general conjectures—especially uniformization in full generality and the optimality of pinching constants—remain central open problems, with partial counterexamples and new analytic obstructions delineating the intricate landscape.