Scientists / Open questions

丘成桐

Shing-Tung Yau

微分几何、几何分析

1982 年菲尔兹奖

57 个问题

The existence/nonexistence boundary for complete Kähler–Einstein metrics on complements of exceptional divisors and higher-codimension subvarieties
Extending text-driven video montage beyond themed text: arbitrary text scope, professional fine-grained control, and scalable cinematography-aware assembly (the Write-a-Video future-work agenda)
Boundary branch points of least-area disks in the classical Plateau problem for smooth non-analytic boundary curves
Integrality and divisibility arithmetic of mirror maps and instanton numbers for Calabi-Yau threefolds: the open core of Lian-Yau (1996)
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
Unconditional positivity of the Bondi mass in four-dimensional Bondi-Sachs radiating spacetimes (the positive mass conjecture at null infinity)
Black hole solutions of the Einstein-Dirac-Maxwell system: eliminating the horizon power-law assumption, settling the extremal case, and extending to stationary axisymmetry
Convergence theory of the conformal energy minimization boundary iteration for disk conformal parameterizations
Foundational open problems for the Wang–Yau quasi-local mass: positivity without admissibility, solvability of the optimal embedding equation, monotonicity, and horizons
Lifting harmonic maps from the symmetric-space quotient back to Griffiths period domains, and harmonic-map finiteness and rigidity for general period mappings
Full proof of the Xu–Yau tree-sum conjectures in Kähler tensor calculus: closed-form characterization of the curvature and pointed tree sets for arbitrary index orderings
Zeroes of the Ricci tensor of the base of Yau's torus-bundle structure theorem for compact complex submanifolds of complex tori
Pólya's conjecture: sharp universal lower bounds for the Dirichlet eigenvalues of general bounded Euclidean domains
Existence of stable bundles with prescribed Chern classes on simply-connected Calabi–Yau threefolds: the Douglas–Reinbacher–Yau conjecture and extremal bundles
Exact string-theoretic status of the stringy cosmic string backgrounds on noncompact Calabi-Yau manifolds: exact N=2 superconformal field theories, persistence beyond the adiabatic truncation, and trapped zero modes
Geometry of the $G_2$ moduli space beyond the Calabi–Yau analogy: geometric meaning of its curvature, the $CY_3\times S^1$ product case and global structure
Unresolved reliability and generality issues of the Yau-Lu unitary-embedding method for computing all eigenvalues of large sparse symmetric matrices
Goal-driven selection and joint optimization of canonical target curvature arrangements in Ricci-flow brain surface conformal parameterization
Convergence theory for computing quasiconformal maps via the auxiliary metric and discrete Yamabe flow beyond disk topology
Perfect tunneling in equilateral triple-well potentials on graphs: the Lin–Lippner–Yau conjecture and the classification of triple-well tunneling
Connecting quantum tunneling from three-dimensional black holes to near-horizon BTZ sectors of higher-dimensional black holes, and the unexplored properties of Sol-asymptotic topological black holes
Sharp rigidity and classification of constant-scalar-curvature hypersurfaces under weak curvature hypotheses and extensions of the Cheng–Yau operator beyond space forms
Generalized Bergman metrics, tautological line bundle negativity over the moduli of Riemann surfaces, and the general existence problem for the Hull–Strominger system (open problems from Yau's 2010 survey 'Metrics on complex manifolds')
Extending the FEM holomorphic-differential and Laurent-series conformal parameterization pipeline from multiply connected genus-zero surfaces to high-genus surfaces

返回开放问题

The existence/nonexistence boundary for complete Kähler–Einstein metrics on complements of exceptional divisors and higher-codimension subvarieties

Extending text-driven video montage beyond themed text: arbitrary text scope, professional fine-grained control, and scalable cinematography-aware assembly (the Write-a-Video future-work agenda)

Boundary branch points of least-area disks in the classical Plateau problem for smooth non-analytic boundary curves

Integrality and divisibility arithmetic of mirror maps and instanton numbers for Calabi-Yau threefolds: the open core of Lian-Yau (1996)

Yau's open questions for positive-curvature surfaces in R3\mathbb{R}^3R3: the n=2n=2n=2 case of the pinched-curvature diameter bounds (Theorem 8) and boundedness of surfaces with constant positive Gaussian curvature

Unconditional positivity of the Bondi mass in four-dimensional Bondi-Sachs radiating spacetimes (the positive mass conjecture at null infinity)

Black hole solutions of the Einstein-Dirac-Maxwell system: eliminating the horizon power-law assumption, settling the extremal case, and extending to stationary axisymmetry

Convergence theory of the conformal energy minimization boundary iteration for disk conformal parameterizations

Foundational open problems for the Wang–Yau quasi-local mass: positivity without admissibility, solvability of the optimal embedding equation, monotonicity, and horizons

Lifting harmonic maps from the symmetric-space quotient back to Griffiths period domains, and harmonic-map finiteness and rigidity for general period mappings

Full proof of the Xu–Yau tree-sum conjectures in Kähler tensor calculus: closed-form characterization of the curvature and pointed tree sets for arbitrary index orderings

Zeroes of the Ricci tensor of the base of Yau's torus-bundle structure theorem for compact complex submanifolds of complex tori

Pólya's conjecture: sharp universal lower bounds for the Dirichlet eigenvalues of general bounded Euclidean domains

Existence of stable bundles with prescribed Chern classes on simply-connected Calabi–Yau threefolds: the Douglas–Reinbacher–Yau conjecture and extremal bundles

Exact string-theoretic status of the stringy cosmic string backgrounds on noncompact Calabi-Yau manifolds: exact N=2 superconformal field theories, persistence beyond the adiabatic truncation, and trapped zero modes

Geometry of the G2G_2G2​ moduli space beyond the Calabi–Yau analogy: geometric meaning of its curvature, the CY3×S1CY_3\times S^1CY3​×S1 product case and global structure

Unresolved reliability and generality issues of the Yau-Lu unitary-embedding method for computing all eigenvalues of large sparse symmetric matrices

Goal-driven selection and joint optimization of canonical target curvature arrangements in Ricci-flow brain surface conformal parameterization

Convergence theory for computing quasiconformal maps via the auxiliary metric and discrete Yamabe flow beyond disk topology

Perfect tunneling in equilateral triple-well potentials on graphs: the Lin–Lippner–Yau conjecture and the classification of triple-well tunneling

Connecting quantum tunneling from three-dimensional black holes to near-horizon BTZ sectors of higher-dimensional black holes, and the unexplored properties of Sol-asymptotic topological black holes

Sharp rigidity and classification of constant-scalar-curvature hypersurfaces under weak curvature hypotheses and extensions of the Cheng–Yau operator beyond space forms

Generalized Bergman metrics, tautological line bundle negativity over the moduli of Riemann surfaces, and the general existence problem for the Hull–Strominger system (open problems from Yau's 2010 survey 'Metrics on complex manifolds')

Extending the FEM holomorphic-differential and Laurent-series conformal parameterization pipeline from multiply connected genus-zero surfaces to high-genus surfaces

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

Geometry of the $G_2$ moduli space beyond the Calabi–Yau analogy: geometric meaning of its curvature, the $CY_3\times S^1$ product case and global structure