Uniformization of complete Kähler manifolds with positive bisectional curvature

1. Transfer initial completeness into local limits and scalar comparison

Proposition 2.1 supplies a strictly psh weight with at most linear growth, finite-jet interpolation, nonvanishing canonical sections, and an exhaustion $u\asymp1+r$ with $|\partial u|_g\leq C$ and $|\partial\bar\partial u|_g\leq C/u$. Tangent-field interpolation uses positivity of $E\otimes\det E$ and weighted $L^2$ solvability; Nakano positivity of $T^{1,0}M$ itself is not assumed. The averaged curvature estimate is $\frac{\int_{B_g(o,L)}R_g\,dV_g}{\operatorname{Vol}_g B_g(o,L)}\leq C/L$ at the fixed center $o$. It supplies neither a center-uniform constant nor maximal volume growth.

Proposition 2.1; §2.1 · pp. 6–8 [Dem82, Theorem 2.6, print p. 467 / PDF p. 12] [Dem82, Theorem 5.1, print p. 475 / PDF p. 20] [CZ05, Theorem 2, p. 2]

On $\{u<L\}$, a conformal blow-up near the boundary produces a complete Hermitian metric $\alpha_L$ with bounded geometry of infinite order. LT20 gives short-time Chern–Ricci flow for each approximation. Potential barriers from canonical sections and a largest-eigenvalue estimate yield two-sided metric bounds on fixed compact sets and time intervals. Chu16 and local higher estimates give a diagonal subsequence converging in local $C^\infty$, including one-sided initial derivatives, with $h_0=g$, $0<h_t\leq g$ and $R_{h_t}\geq0$. Completeness of the approximations alone is not used to infer completeness of the limit.

Theorem 3.1; §§3.1–3.3 · pp. 12–16 [LT20, Definition 2.1; Lemma 2.1, p. 6] [Chu16, Theorem 5.1 and hypotheses H1–H3, pp. 14–15]

The volume loss $\rho=\log(\det g/\det h_t)$ satisfies $\frac{\int_{B_g(o,L)}\rho\,dV_g}{\operatorname{Vol}_g B_g(o,L)}\leq C_TL$ on finite time intervals. An integrable weight $\chi\asymp u^{-k}$ adapted to $g-h_t$ satisfies $\partial\bar\partial\chi\leq C_T\chi h_t$. Together with the cofactor bound $|\operatorname{tr}_{h_t}B|dV_{h_t}\leq C_n|B|_g dV_g$, it removes the error at infinity. Proposition 3.3 proves, for upper semicontinuous viscosity subsolutions with $0\leq v\leq\rho$, $$ \partial_tv\leq e^{-v}\Delta_{h_t}v+(1-e^{-v})R_{h_t}\quad\Longrightarrow\quad v=0. $$ For its smooth majorant $w$, the identity $\partial_t[(e^w-1)dV_{h_t}]=(\Delta_{h_t}w)dV_{h_t}$ closes the weighted-mass Gronwall argument.

Proposition 3.3; proof · pp. 18–20 Lemma 3.4; §3.6 · pp. 19–22

2. Joint positivity yields Harnack inequalities and completeness

Once Theorem 6.3 proves that $N(x,\xi,e^{w+\bar w})$ is psh, the spatial and temporal curvature blocks give nonnegative bisectional curvature and matrix/scalar Harnack inequalities. Initial strict positivity gives positive Ricci curvature on fixed compact sets at small times, and the matrix inequality propagates it. The scalar inequality is integrated along piecewise smooth paths approximating the distance; this precedes completeness and requires no global minimizing geodesic.

Theorem 6.3 · p. 48 Proposition 7.1; proof · pp. 49–50

The least exit length $\ell_L(t)$ from the initial ball $B_g(o,L)$ then satisfies $1+\ell_L(t)\geq(1+L)e^{-2C_T\sqrt t}$. Letting $L\to\infty$ proves completeness of every positive-time slice. Here $C_T$ depends on the finite time interval and the fixed initial geometry.

3. Spread volume contraction to every direction and every compact set

Set $\tau=\log t$, $s=\rho(o,t)$ and $q(s)=ds/d\tau=tR(o,t)$. If $B,A$ are the greatest and least initial-metric singular lengths of a local chart, Lemma 7.3 gives $B\leq C(1+A)^C$. Thus determinant loss forces contraction in every direction. Its constants depend on the fixed compact set, chart radius, lower metric bound and initial geometry in a neighborhood, independently of the particular chart's total distortion. Uniformly on each fixed compact, as $s\to\infty$, $$ \rho(x,t)/s\longrightarrow1,\qquad e^{-2s}g\leq h_t\leq e^{-c_Ks}g. $$

Lemma 7.3; proof · pp. 52–54 Propositions 7.4–7.5; proofs · pp. 55–58

On selected Ricci-pinched windows, rescaled flows pulled back to tangent-space balls have local $C^\infty$ subsequential limits. Equality in the limiting Harnack inequality forces the real Hessian of scalar curvature to be strictly negative. This estimate transfers back to small intrinsic balls in the original manifolds. The scalar-curvature gradient flow contracts each fixed loop, proving simple connectivity. Neither a global noncollapsed limit nor global injectivity of the local charts is assumed.

Simple connectivity allows coherent continuation of inverse charts $f_s$ over an exhaustion. The initial bound $\sup_K|f_s|\leq C_Ke^{-a_Ks}$ is improved using local $L^1$ psh subsequential limits of $s^{-1}\log|f_s|$ and the Liouville principle of Lemma 7.6. This yields one $\sigma>0$ for all fixed compacts: $\sup_K|f_s|\leq e^{-\sigma s}$, with a starting threshold allowed to depend on $K$. Transitions $F_{s,u}$ retain $|\det F'_{s,u}(0)|=e^{-(u-s)/2}$ and jet/small-ball bounds uniform in the later endpoint $u$.

Proposition 7.7; proof · pp. 59–61

4. Account for nonuniform contraction through forward degrees

For a fixed jet order $d$, Proposition 8.1 constructs local coordinate changes $H_k$ and polynomial automorphisms $G_k$ satisfying $J_d(H_{k+1}F_k)=J_d(G_kH_k)$. It first removes the nonconstant Jacobian jet, then realizes divergence-free homogeneous fields by shears. The nonlinear shears have Jacobian one; the full $G_k$ retains the constant determinant of its contracting linear part. For $n=1$, divergence-free homogeneous fields of degree at least two vanish, so the construction also covers that case.

Proposition 8.1 · p. 62 Lemma 8.2 · p. 63 [FL14, Proof of Theorem 3.1 (divergence-free shears), pp. 8–9]

The manuscript preserves weighted component degrees on regular intervals and charges degree growth on bad intervals to growth of $q$. Since $\liminf_{s\to\infty}q(s)/s\leq2$, the forward compositions $G_{0,k}=G_{k-1}\circ\cdots\circ G_0$ satisfy $\deg G_{0,k}\leq Ds_k$ along an unbounded subsequence. Proposition 9.2 turns the corrected-coordinate limit into a biholomorphism $\Psi:M\to\Omega\subset\mathbb C^n$. At this stage a proper image remains possible.

Lemma 8.4; proof · pp. 68–69 Proposition 9.2 · p. 71

5. Surjectivity follows by contradicting a maximal ball radius

Take $0<\gamma'<\sigma/2$ and suppose $R_* =\sup\{R:B(R)\subset\Omega\}<\infty$. For the degree-controlled subsequence choose $0<R_1<R_*<R_2$ with $D\log(R_2/R_1)<\sigma/2-\gamma'$. Since $\overline{B(R_1)}\Subset\Omega$, Proposition 9.2 and polynomial growth give $$ \sup_{B(R_2)}|G_{0,k}| \leq (R_2/R_1)^{Ds_k}e^{-(\sigma/2)s_k} <e^{-\gamma's_k}. $$ The same proposition gives $G_{0,k}^{-1}(B(e^{-\gamma's_k}))\subset\Omega$, hence $B(R_2)\subset\Omega$, a contradiction. Therefore $\Omega=\mathbb C^n$. Together with injectivity and nonsingularity, this proves the asserted biholomorphism, and hence Steinness and contractibility. This final step needs neither inverse-composition degree bounds nor a uniform instantaneous contraction ratio.

§9.4; proof of Theorem 1.1 · pp. 73–74

1. Obtain boundary estimates that survive removal of the area penalty

Initial dual-norm positivity and $h_t\leq g$ give $N(\cdot,0)\leq v\leq N(\cdot,t)$. The disc infimum need not be attained. Boundary walls and a penalty $\varepsilon\mathcal E$ confine disc interiors to a compact set for fixed wall index $j$ and $\varepsilon>0$. That interior compact may depend on $\varepsilon$, but the coefficient in the final deformation error $C_j\varepsilon(1+\mathcal E)$ must depend only on boundary data. The minimizing-sequence limit, $\varepsilon\downarrow0$, and the subsequent $j\to\infty$ are distinct steps.

§4; Lemma 4.1 · pp. 22–23 Lemma 5.2; §§5.3–5.5 · pp. 34–41

After removing the negative part of the boundary Hessian, a positive majorant is factored as $H^*H$. The holomorphic invertible factors smooth up to the boundary supplied by BR03 permit disc-dependent test directions. Clock deformations run backward to remain admissible even when boundary times approach $T$. For a lower-test Hessian $W=\bigl(\begin{smallmatrix}A&B\\B^*&C\end{smallmatrix}\bigr)$ with $C>0$, Proposition 5.1 supplies $K\geq0$, $K\geq A-BC^{-1}B^*$ and $\partial_t\phi\geq\operatorname{tr}(C^{\mathsf T}K)$. This is not an assertion that the horizontal Hessian itself is nonnegative.

[BR03, Theorem 2.1, print p. 887 / PDF p. 4] Proposition 5.1 · pp. 31–32

2. Sum common-base contacts to obtain the determinant inequality

In each cotangent fiber maximize $\log\det B$ among $B>0$ with $\xi^*B\xi\leq v(x,\xi,t)$. Set $\mathcal L=\log\det B$, $b_t=h_t^{-\mathsf T}$ and $F_0=\log\det b_t-\mathcal L$. Then $0\leq F_0\leq\rho$. In coordinates where $B=I$, contact vectors satisfy $\sum_i\alpha_i\xi_i\xi_i^*=I$; replicating the list makes $\max_i\alpha_i$ small.

A product support and inf-convolution select simultaneous contacts with one shared $x,t$. The vertical Levi blocks remain positive, allowing Proposition 5.1 and summation of Schur complements. The temporary decrease of the test Hessian is removed at each fixed contact before removing regularization and finally passing to small contact weights. Proposition 6.2 yields $\partial_t\mathcal L\geq e^{-F_0}\Delta_{h_t}\mathcal L$ in the lower viscosity sense.

Lemma 6.1; Proposition 6.2 · pp. 42–47

3. Return from zero determinant deficit to the disc inequality

The function $\ell=\log\det b_t$ satisfies $\partial_t\ell=\Delta_{h_t}\ell=R_{h_t}$. Thus $F_0$ satisfies the upper viscosity inequality of Proposition 3.3, giving $F_0=0$. Since $B\leq b_t$ and $\det B=\det b_t$, we have $B=b_t$, hence $N\leq v\leq N$. Also $v\leq v^{\rm raw}\leq N$, so the raw disc infimum equals $N$ and every admissible disc satisfies the submean inequality. Arbitrary finite horizons give joint positivity including complexified logarithmic time.

Proposition 3.3; proof · pp. 18–20 Theorem 6.3 · p. 48

1. Sum jet errors inside the valid domains of backward iteration

The manuscript chooses $\nu>0$ from the contraction exponents, then $d$ with $(d+1)\nu>30$, then a small macro width $h$, and finally a late starting time. This order uses independence from $h$ of the coefficient $C_d$ multiplying $\Delta s_k$. For $\zeta_k=H_k(z_k)$ the defect $r_k=\zeta_{k+1}-G_k\zeta_k$ is bounded by $e^{-10s_k}$. The Lipschitz bound for $G_k^{-1}$ is used only inside its prescribed $B(e^{-\nu s_{k+1}})$.

§9.1, (9.1)–(9.7) · pp. 69–70

Backward induction proves the error bound and containment of the comparison segments in those balls simultaneously. The sum $\sum_{i\geq k}e^{-10s_i+s_{i+1}-s_k}\leq e^{-6s_k}$ closes the argument. Thus $\xi_k=\lim_{l\to\infty}G_{k,l}^{-1}\zeta_l$ converges locally uniformly and satisfies $\xi_{k+1}=G_k\xi_k$. Two regimes are used: $z_k=f_{s_k}(x)$ on fixed compact sets and $z_k=F_{s_a,s_k}(z)$ from shrinking chart balls. The latter is needed for ball inclusion.

Lemma 9.1 · p. 71

2. Prove injectivity and nonsingularity as well as holomorphic convergence

Coherent inverse charts and $\xi_{k+1}=G_k\xi_k$ patch the local limits. If two points have the same image, their raw coordinates differ by at most $e^{-5s_k}$. Comparing with $h_{t(s_k)}\geq e^{-2s_k}g$ produces a path exiting a fixed initial-metric ball with length tending to zero, a contradiction.

Nonsingularity does not follow merely from locally uniform convergence. The lower bound $cs_k^{-C}e^{-s_k}$ for $d(H_kf_{s_k})$ is compared with the differential error obtained by applying Cauchy estimates to the $e^{-6s_k}$ error on a slightly larger fixed neighborhood. Hence $d\Psi$ is nonsingular. Injectivity and the local inverse theorem show that $\Psi:M\to\Omega$ is biholomorphic. Applying chart contraction on $\Psi^{-1}(K)$ yields forward smallness on each compact subset of the image.

§9.3, (9.10)–(9.14) · pp. 71–73

3. Produce a ball contained in the transformed image

On $B(e^{-\gamma s_k})$, compare $T_k=G_{0,k}\Psi\Phi_{s_k}$ with the linear part $A_k=H_k'(0)$. Polynomial bounds for $A_k^{-1}$ and Cauchy estimates make the differential of $A_k^{-1}(T_k-A_k)$ small. Since $\gamma'>\gamma$, for $|w|\leq e^{-\gamma's_k}$ the equation $z=A_k^{-1}w-A_k^{-1}(T_k(z)-A_kz)$ is solved by a contraction on a small ball. Thus $B(e^{-\gamma's_k})\subset G_{0,k}(\Omega)$. This inclusion and forward smallness apply to the same $G_{0,k}$ in the final step of the main theorem.

§9.3, (9.10)–(9.14) · pp. 71–73