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An L² Einstein Gap for Nonnegatively Curved Four-Manifolds
expertly designed by an internal OpenAI model · released 2026-10-05
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IntroductionFor a Riemannian four-manifold \((M,g)\), let \[E_g=\mathop{\mathrm{Ric}}_g-\frac{\mathop{\mathrm{Scal}}_g}{4}g, \qquad \mathcal D(g)=\int_M |E_g|_g^2\,d\mu_g.\] The tensor \(E_g\) vanishes precisely when \(g\) is Einstein, provided that \(M\) is connected. The energy \(\mathcal D\) is unchanged by constant rescaling in dimension four. It is therefore natural to ask whether a classification of Einstein metrics remains valid at the level of smooth manifolds when \(\mathcal D\) is small. We address this question under nonnegative sectional curvature. We take the exact Einstein classification in (OpenAI 2026, Corollary 1.2) as the following premise. Classification Assumption 1. Every connected smooth closed Riemannian four-manifold \((N,h)\) satisfying \[\mathop{\mathrm{Ric}}_h=3h,\qquad \sec_h\ge0\] has universal Riemannian cover isometric to one of the following: the round \(S^4\) of sectional curvature \(1\); Fubini–Study \(\mathbb{CP}^2\) normalized by \(\mathop{\mathrm{Ric}}=3g\); or \(S^2(1/\sqrt3)\times S^2(1/\sqrt3)\) with its product metric. This premise is used only for an exactly Einstein metric; no stability statement is assumed. Theorem 2. Assume Classification Assumption 1. There is a universal constant \(\varepsilon_0>0\) with the following property. If \((M,g)\) is a connected, simply connected, smooth closed Riemannian four-manifold with \(\sec_g\ge0\) and \[\int_M\left|\mathop{\mathrm{Ric}}_g-\frac{\mathop{\mathrm{Scal}}_g}{4}g\right|_g^2\,d\mu_g <\varepsilon_0,\] then \(M\) is diffeomorphic, without a prescribed orientation, to the standard smooth \(S^4\), \(\mathbb{CP}^2\), or \(S^2\times S^2\). There are no auxiliary bounds on curvature, volume, diameter, injectivity radius, Sobolev constants, or curvature derivatives. The conclusion is about the underlying smooth manifold, rather than the original metric. This distinction matters because small \(L^2\) error does not itself give pointwise curvature control. Context and methodsThe exact Einstein problem has been studied under several additional hypotheses. Gursky and LeBrun obtained the Fubini–Study conclusion for nonnegative sectional curvature and a nonzero positive-definite intersection form (Gursky and LeBrun 1999). Cao and Tran established sectional-pinching criteria (Cao and Tran 2018), and Liu classified the \(T^2\)-invariant case on closed simply connected four-manifolds (Liu 2025). Here the classification premise is supplied by Zero-Plane Rigidity for Einstein Four-Manifolds (OpenAI 2026, Corollary 1.2). Our problem is instead to transfer that exact classification to metrics with small critical Ricci error. The relation between Ricci curvature, scalar curvature, and topology is especially strong in dimension four: the Chern–Gauss–Bonnet formula controls the Euler characteristic by the total squared curvature. The almost-Schur inequality of De Lellis and Topping (De Lellis and Topping 2012) controls the variance of scalar curvature by the trace-free Ricci tensor under nonnegative Ricci curvature. We use its Bianchi–Bochner argument to choose a scale in which the Ricci tensor is close in \(L^2\) to \(3g\). Integral versions of Myers’ diameter argument combine second variation with control of curvature along many geodesics. The segment inequality of Cheeger and Colding (Cheeger and Colding 1996) and the integral-curvature diameter estimates of Petersen and Sprouse (Petersen and Sprouse 1998; Sprouse 2000) provide the relevant context. We give the short segment-averaging argument needed here, together with an elementary derivation of the initial Sobolev inequality. These estimates prevent collapse before any smoothing is performed. Hamilton’s Ricci flow (Hamilton 1982) supplies the regularization. Perelman’s entropy monotonicity (Perelman 2002) propagates a scalar-curvature-weighted Sobolev inequality. The passage from entropy to Sobolev inequalities along Ricci flow is developed by Ye and Zhang (Ye 2015; Zhang 2007); we include the form needed here, with constants that are uniform even before a common existence interval is known. Shi’s derivative estimates (Shi 1989) and Hamilton’s compactness theorem (Hamilton 1995) then turn curvature control at positive times into smooth convergence. Smoothing from small critical curvature concentration was developed, in particular, by Chan, Chen, and Lee (Chan et al. 2022). The formulation used here isolates a globally small high part of the curvature. Consider a sequence \((M_j,g_j)\) satisfying the geometric hypotheses of Theorem 2 with \(\mathcal D(g_j)\to0\). After normalization, nonnegative sectional curvature gives, for a fixed threshold \(K_0\), \[\left\lVert (|\mathop{\mathrm{Rm}}_{g_j}|-K_0)_+\right\rVert_{L^2(g_j)}\longrightarrow0.\] A moving threshold propagates this estimate. It yields both a uniform Sobolev inequality and noncollapse along each existing flow; a curvature-concentration contradiction then establishes a common lifespan. The same tail bound controls the evolving Einstein error. Thus the high-part formulation supplies both the regularization and the defect estimate needed for the gap argument. A second point concerns sectional curvature. Its nonnegativity is not assumed to persist under the flow. Instead, we bound the integral of its negative part by \(Ct\). The limit is an exactly Einstein shrinking flow, and its explicit scaling converts this weak defect estimate into nonnegative sectional curvature of the fixed Einstein metric. This gives the classification-independent compactness statement of Proposition 12; classification then identifies the limiting manifold. Proof organizationSection 2 normalizes the sequence and establishes the initial geometric and Sobolev bounds. Section 3 obtains a uniform smoothing interval by controlling the high-curvature part. Section 4 propagates the Einstein error and estimates the negative sectional curvature. Section 5 constructs the compact Einstein limit, identifies its sectional curvature, and proves Theorem 2. All manifolds are smooth, and closed means compact without boundary. We use \(R=\mathop{\mathrm{Scal}}\) and the curvature convention in which \(\mathop{\mathrm{Rm}}(X,Y,X,Y)\) is the sectional curvature of an orthonormal pair. The Laplacian is \(\Delta=\operatorname{tr}\nabla^2\). Norms and integrals use the displayed metric, or the current metric along a flow, and are not volume-normalized. Constants denoted by \(C\) or \(c\) may change from line to line. Uniformity always means independence of the sequence index, after discarding finitely many terms. Normalization and initial geometryThroughout the proof let \((M_j,g_j)\) be connected, simply connected, closed four-manifolds with \[ \sec_{g_j}\ge0,\qquad \left\lVert E_{g_j}\right\rVert_2\longrightarrow0. \tag{1}\] The aim of this section is to obtain uniform geometric bounds without assuming a curvature maximum bound. We suppress \(j\) when considering one metric in the sequence. Proposition 3. After constant rescaling, the metrics in (1) satisfy \[ \frac1V\int_M R\,d\mu=12,\qquad \left\lVert R-12\right\rVert_2\le4\left\lVert E\right\rVert_2,\qquad \left\lVert \mathop{\mathrm{Ric}}-3g\right\rVert_2\longrightarrow0, \tag{2}\] where \(V=\mathop{\mathrm{Vol}}(M,g)\). There are uniform positive constants \(v_0,V_0,D_0\) such that \[ v_0\le V\le V_0,\qquad \mathop{\mathrm{diam}}(M,g)\le D_0. \tag{3}\] Proof. Simple connectivity implies orientability and \(b_1=0\). Poincaré duality therefore gives \(\chi(M)=2+b_2(M)\ge2\). By Chern–Gauss–Bonnet, \[ 2\le\chi(M)\le C\int_M|\mathop{\mathrm{Rm}}|^2\,d\mu. \tag{4}\] On the space of algebraic curvature tensors, the maximum absolute sectional curvature is a norm: sectional curvatures determine the tensor. Equivalence of norms in this finite-dimensional space, together with \(0\le\sec\le R/2\), gives a dimensional constant \(C_0\) such that \[ |\mathop{\mathrm{Rm}}|\le C_0R. \tag{5}\] The average scalar curvature is positive, since otherwise \(R\equiv0\) and (5) contradicts (4). Rescale so that its average is \(12\). The \(L^2\) norm of \(E\) is unchanged. For completeness, the almost-Schur argument (De Lellis and Topping 2012, sec. 2.1) in the form needed here is as follows. Solve \(\Delta f=R-12\). The contracted Bianchi identity says \(\mathop{\mathrm{div}}E=\tfrac14dR\), so integration by parts gives \[\int_M(R-12)^2\,d\mu=4\int_M\langle E,\nabla^2f\rangle\,d\mu.\] Since \(\mathop{\mathrm{Ric}}\ge0\), the integrated Bochner formula yields \(\left\lVert \nabla^2f\right\rVert_2\le\left\lVert \Delta f\right\rVert_2\). Cauchy–Schwarz proves the second assertion in (2). Orthogonality of the trace and trace-free parts gives \[ \left\lVert \mathop{\mathrm{Ric}}-3g\right\rVert_2^2 =\left\lVert E\right\rVert_2^2+\frac14\left\lVert R-12\right\rVert_2^2\longrightarrow0. \tag{6}\] Moreover, \[\int_MR^2\,d\mu=144V+\left\lVert R-12\right\rVert_2^2.\] Combining this identity with (4) and (5) gives \(V\ge v_0>0\) for all sufficiently large \(j\). It remains to control the diameter. Write \(D=\mathop{\mathrm{diam}}(M,g)\) and choose points realizing it. Let \(A_1,A_2\) be the balls of radius \(D/4\) about these points. Bishop–Gromov comparison gives \(\mathop{\mathrm{Vol}}(A_i)\ge8^{-4}V\), by comparison with the ball of radius \(2D\). For any nonnegative continuous function \(H\), the segment inequality is \[ \int_{A_1\times A_2}\int_{\gamma_{xy}}H\,ds\,d\mu(x)d\mu(y) \le CD\bigl(\mathop{\mathrm{Vol}}(A_1)+\mathop{\mathrm{Vol}}(A_2)\bigr)\int_MH\,d\mu, \tag{7}\] where \(\gamma_{xy}\) is the minimizing geodesic, unique for almost every pair. This is the nonnegative-Ricci case of (Cheeger and Colding 1996, Theorem 2.11). To see the estimate directly, fix \(x\) and use polar coordinates \(y=\exp_x(r\theta)\) before cut time. On the half-segment \(r/2\le s\le r\), Jacobian comparison gives \[J_x(r,\theta)\le(r/s)^3J_x(s,\theta)\le8J_x(s,\theta).\] Changing the order of integration bounds the integral over this half, and over \(y\), by \(CD\int_MH\,d\mu\). Integrating in \(x\in A_1\) gives its contribution to (7). Reverse the geodesic and fix \(y\) to treat the other half. The cut locus has measure zero. Apply (7) to \(H=|\mathop{\mathrm{Ric}}-3g|\). Some minimizing geodesic between the two balls has length \(\ell\ge D/2\) and satisfies \[ \int_0^\ell H(\gamma(s))\,ds\le\delta_jD, \qquad \delta_j=CV^{-1/2}\left\lVert \mathop{\mathrm{Ric}}-3g\right\rVert_2\longrightarrow0. \tag{8}\] Use three parallel orthonormal normal fields along \(\gamma\), each multiplied by \(\sin(\pi s/\ell)\), in the index form. Summing gives \[\frac{3\pi^2}{2\ell} \ge\int_0^\ell\sin^2(\pi s/\ell) \mathop{\mathrm{Ric}}(\dot\gamma,\dot\gamma)\,ds \ge\frac{3\ell}{2}-\delta_jD.\] Because \(D\le2\ell\), we obtain \((3/2-2\delta_j)\ell^2\le3\pi^2/2\). Thus \(D\) is uniformly bounded. Bishop comparison now gives the uniform upper volume bound as well. ◻ A support estimate and the Sobolev inequalityWe will need Sobolev inequalities both initially and along the flow. The following elementary truncation lemma allows us to derive both from estimates for functions with small support. Lemma 4. Let \((M,g)\) be a closed four-manifold, \(a>0\), and \(P\ge0\) a smooth function. Set \[\mathcal E(f)=\int_M(a|\nabla f|^2+Pf^2)\,d\mu.\] Suppose that for every Lipschitz function \(f\), with \(m=\mathop{\mathrm{Vol}}\{f\ne0\}\), one has \[ \left\lVert f\right\rVert_2^2\le C m^{1/2}\mathcal E(f). \tag{9}\] Then \(\left\lVert f\right\rVert_4^2\le C'\mathcal E(f)\) for every such \(f\), where \(C'\) depends only on \(C\). Proof. For \(k\in\mathbb Z\) define \[m_k=\mathop{\mathrm{Vol}}\{|f|>2^k\},\qquad f_k=\min\{(|f|-2^k)_+,2^k\}.\] The assumed estimate implies \[2^{2k}m_{k+1}\le\left\lVert f_k\right\rVert_2^2 \le C m_k^{1/2}\mathcal E(f_k).\] The gradients of \(f_k\) occupy disjoint layers up to null sets, and \(\sum_k f_k^2\le C_1f^2\) pointwise by a geometric series. Consequently \(\sum_k\mathcal E(f_k)\le C_2\mathcal E(f)\), with numerical constants. Put \(b_k=2^{4k}m_k\). The preceding estimate becomes \[b_{k+1}\le16C b_k^{1/2}\mathcal E(f_k).\] The sum \(B=\sum_kb_k\) is finite and comparable, with numerical constants, to \(\int_M|f|^4\,d\mu\): integrate the geometric series \(\sum_{2^k<|f|}2^{4k}\) pointwise. Summing over \(k\) and using \(b_k^{1/2}\le B^{1/2}\) gives \(B\le16CC_2B^{1/2}\mathcal E(f)\). Division by \(B^{1/2}\) when \(B>0\) proves the result; the case \(B=0\) is immediate. ◻ Proposition 5. The normalized metrics of Proposition 3 satisfy \[ \left\lVert f\right\rVert_4^2\le S\int_M(|\nabla f|^2+f^2)\,d\mu \tag{10}\] for all Lipschitz functions \(f\), with a uniform constant \(S\). Proof. We establish (9) with \(a=P=1\). Let \(\bar f=V^{-1}\int_Mf\,d\mu\). Polar integration of \(|f(x)-f(y)|\) along minimizing segments yields \[ |f(x)-\bar f| \le\frac{CD^4}{V}\int_M\frac{|\nabla f(y)|}{d(x,y)^3}\,d\mu(y). \tag{11}\] Indeed, on a ray from \(x\), the coefficient of \(|\nabla f|\) at distance \(s\) is bounded using \[\int_s^{\operatorname{cut}(\theta)}J_x(r,\theta)\,dr \le \frac{J_x(s,\theta)}{s^3}\int_s^D r^3\,dr \le\frac{D^4J_x(s,\theta)}{4s^3}.\] This proves (11); absolute continuity along almost every ray justifies it for Lipschitz functions. For any measurable set \(A\) of volume \(m>0\), comparison with Euclidean polar measure gives \[\sup_x\int_A d(x,y)^{-3}\,d\mu(y)\le C m^{1/4}.\] To verify this, split the integral at radius \(\rho=m^{1/4}\). The inner integral is at most \(C\rho\) because \(J_x(r,\theta)\le r^3\); the outer integral is at most \(m\rho^{-3}\). By symmetry the same bound holds with \(x\) and \(y\) interchanged. The integral operator with this kernel, restricted to \(A\) in both variables, therefore has \(L^2\) norm at most \(Cm^{1/4}\) by the Schur estimate. Take \(A=\{f\ne0\}\). The gradient vanishes almost everywhere off \(A\), and (3) bounds \(D^4/V\). Taking \(L^2(A)\) norms in (11) gives \[\left\lVert f\right\rVert_2\le Cm^{1/4}\left\lVert \nabla f\right\rVert_2+\sqrt m\,|\bar f| \le Cm^{1/4}\left\lVert \nabla f\right\rVert_2+\frac mV\left\lVert f\right\rVert_2.\] If \(m/V\le1/2\), absorb the last term and square. If \(m/V>1/2\), then \(m\ge v_0/2\), and (9) follows directly from the \(\int f^2\) term in the energy. Lemma 4 completes the proof. ◻ A uniform smoothing intervalLet \(g_j(t)\) be the maximal smooth unnormalized Ricci flows with the normalized initial metrics above: \[\partial_tg_j=-2\mathop{\mathrm{Ric}}_{g_j},\qquad g_j(0)=g_j.\] Short-time existence and the curvature continuation criterion are standard on closed manifolds (Hamilton 1982). Initially each flow has its own maximal time \(\mathcal T_j>0\). Until a common lower bound for these times is proved, every estimate refers only to existing times. The scalar evolution and volume variation are \[ \partial_tR=\Delta R+2|\mathop{\mathrm{Ric}}|^2,\qquad \partial_td\mu=-R\,d\mu. \tag{12}\] Thus \(R\ge0\) and \(\mathop{\mathrm{Vol}}(M,g_j(t))\le V_0\) throughout smooth existence. We first propagate a Sobolev inequality with a scalar-curvature weight, then use smallness of the high-curvature part to remove that weight. Entropy supplies a weighted Sobolev inequalityProposition 6. There is a uniform \(S_1\) such that, whenever \(0\le t<\min\{1,\mathcal T_j\}\), \[ \left\lVert f\right\rVert_4^2\le S_1\int_M\bigl(4|\nabla f|^2+(R+1)f^2\bigr)\,d\mu \tag{13}\] for every Lipschitz function \(f\) on \((M_j,g_j(t))\). Proof. In dimension four, Perelman’s entropy is \[\begin{align*} \mu(g,\tau)=\inf_{\int f^2\,d\mu=1}\bigg\{ &\tau\int_M(4|\nabla f|^2+Rf^2)\,d\mu -\int_Mf^2\log f^2\,d\mu\\ &-2\log(4\pi\tau)-4\bigg\}, \end{align*}\] where the infimum is over smooth positive \(f\). Its monotonicity gives \[ \mu(g_j(t),\tau)\ge\mu(g_j(0),\tau+t),\qquad \tau>0 \tag{14}\] (Perelman 2002, sec. 3). Here \(\tau+t\) is an entropy parameter, not a later flow time, so (14) imposes no additional lifespan requirement. At time zero, Jensen’s inequality and (10) give \[\int_Mf^2\log f^2\,d\mu\le\log\int_Mf^4\,d\mu \le2\log\bigl(S(y+1)\bigr),\qquad y=\int_M|\nabla f|^2\,d\mu,\] for \(\int f^2\,d\mu=1\). Since \(R\ge0\), \[\mu(g_j(0),\tau)\ge \inf_{y\ge0}\{4\tau y-2\log(\tau(y+1))\}-C\ge-C' \quad(0<\tau\le2).\] The last bound follows by writing \(z=\tau(y+1)\): the expression in braces is \(4z-2\log z-4\tau\), uniformly bounded below in this range. Consequently (14) gives, at all times under consideration, \[ \int_Mf^2\log f^2\,d\mu \le\tau\mathcal E_0(f)-2\log\tau+C, \qquad \mathcal E_0(f)=\int_M(4|\nabla f|^2+Rf^2)\,d\mu, \tag{15}\] whenever \(\left\lVert f\right\rVert_2=1\) and \(0<\tau\le1\). Approximation of \(|f|\) by positive smooth functions extends this to Lipschitz \(f\), with \(0\log0=0\). Let \(m=\mathop{\mathrm{Vol}}\{f\ne0\}\). For a normalized \(f\), Jensen’s inequality on its support gives \(\int f^2\log f^2\,d\mu\ge\log(1/m)\). Choose \(\tau=(1+\mathcal E_0(f))^{-1}\) in (15). It follows that \[\log(1/m)\le2\log(1+\mathcal E_0(f))+C.\] Exponentiation, followed by homogeneity for arbitrary \(f\), yields \[\left\lVert f\right\rVert_2^2\le Cm^{1/2}\left(\mathcal E_0(f)+\left\lVert f\right\rVert_2^2\right).\] Apply Lemma 4 with \(a=4\) and \(P=R+1\). ◻ The high-curvature part stays smallIn orthonormal frames evolving by \(\partial_te_i=\mathop{\mathrm{Ric}}^\sharp e_i\), the curvature equation has the form \[ \partial_t\mathop{\mathrm{Rm}}=\Delta\mathop{\mathrm{Rm}}+\mathcal Q(\mathop{\mathrm{Rm}}),\qquad |\mathcal Q(\mathop{\mathrm{Rm}})|\le C_q|\mathop{\mathrm{Rm}}|^2. \tag{16}\] Here time derivatives are derivatives of the moving-frame components; the connection Laplacian has the sign fixed in Section 1. It follows that \(u=|\mathop{\mathrm{Rm}}|\) satisfies \((\partial_t-\Delta)u\le C_qu^2\) wherever \(u>0\). Choose the scalar function \(K\) by \[ K(0)=12C_0,\qquad K'=C_qK^2, \tag{17}\] where \(C_0\) is from (5). Fix \(T\in(0,1/6)\) so small that \(K\) is bounded on \([0,T]\), and set \[w_j(t)=\bigl(|\mathop{\mathrm{Rm}}_{g_j(t)}|-K(t)\bigr)_+.\] At time zero, (5) implies \[ 0\le w_j(0)\le C_0(R_{g_j}-12)_+,\qquad \left\lVert w_j(0)\right\rVert_2\longrightarrow0. \tag{18}\] Proposition 7. After discarding finitely many indices, there are uniform constants \(S_2,C\) such that, for \(0\le t<\min\{T,\mathcal T_j\}\), \[\begin{align*} \left\lVert f\right\rVert_4^2&\le S_2\int_M(|\nabla f|^2+f^2)\,d\mu, \tag{19}\\ \int_M w_j(t)^2\,d\mu&\le e^{Ct}\int_M w_j(0)^2\,d\mu_{g_j}, \tag{20}\\ \int_M|\mathop{\mathrm{Rm}}_{g_j(t)}|^2\,d\mu&\le C. \tag{21}\end{align*}\] The Sobolev estimate holds for every Lipschitz function \(f\). In particular, the supremum of \(\left\lVert w_j(t)\right\rVert_2\) over these existing times tends to zero. Proof. We first work on any interval where \(\left\lVert w\right\rVert_2\le\eta\) for a small fixed \(\eta>0\). Since \(R\le C|\mathop{\mathrm{Rm}}|\le C(K+w)\), the last term in (13) is bounded by a fixed multiple of \(\int f^2\) plus \[C\int_Mwf^2\,d\mu\le C\eta\left\lVert f\right\rVert_4^2.\] Choose \(\eta\) so that this term is absorbed into the left side. This gives (19) with a fixed \(S_2\); decreasing \(\eta\) further does not change the chosen \(S_2\). On \(\{u>K\}\), the equation for \(K\) removes the bounded-background reaction: \[(\partial_t-\Delta)(u-K)\le C_q(u^2-K^2) =C_q(w^2+2Kw).\] Integration by parts and (12) therefore give \[ \frac d{dt}\int_Mw^2\,d\mu \le-2\int_M|\nabla w|^2\,d\mu +2C_q\int_Mw^3\,d\mu+4C_qK\int_Mw^2\,d\mu. \tag{22}\] There is no adverse volume term, since \(-\int Rw^2\,d\mu\le0\). The calculation is valid for the positive part: \(K>0\), so \(u\) is smooth near \(\{u\ge K\}\); the function \((u-K)_+^2\) is continuously differentiable, and Lipschitz truncation justifies integration by parts. Hölder’s inequality and (19) give \[\int_Mw^3\,d\mu\le\eta\left\lVert w\right\rVert_4^2 \le\eta S_2\int_M(|\nabla w|^2+w^2)\,d\mu.\] Decrease \(\eta\) so that \(2C_q\eta S_2\le1\). Then \(Y(t)=\int w^2\,d\mu\) satisfies \(Y'\le CY\) as long as \(Y^{1/2}\le\eta\). By (18), for large \(j\) we have \(Y(0)e^{CT}<\eta^2\). Gronwall’s inequality and continuity exclude a first time with \(Y^{1/2}=\eta\) before \(\min\{T,\mathcal T_j\}\). This proves (19) and (20) on the entire existing interval. Finally, \(u\le K+w\) and \(\mathop{\mathrm{Vol}}(M,g_j(t))\le V_0\) give (21). ◻ Noncollapse and the common lifespanNo common existence time has been used in this argument. We now use its two outputs, uniform Sobolev control and a vanishing curvature tail, to establish one. Lemma 8. There is a uniform \(v_*>0\) such that at every time covered by Proposition 7, \[ \mathop{\mathrm{Vol}}_{g_j(t)}B_{g_j(t)}(x,r)\ge v_*r^4 \quad\text{for all }x\in M_j,\quad 0<r\le1. \tag{23}\] Proof. Write \(V_x(r)=\mathop{\mathrm{Vol}}B(x,r)\) on one smooth slice. Insert into (19) the distance cutoff that is \(1\) on \(B(x,r/2)\), vanishes outside \(B(x,r)\), and has gradient at most \(2/r\). Then \[V_x(r/2)^{1/2}\le Cr^{-2}V_x(r).\] For \(a(r)=V_x(r)/r^4\), this gives \(a(r)\ge c\,a(r/2)^{1/2}\). Iteration yields \[a(r)\ge c^{2(1-2^{-k})}a(2^{-k}r)^{2^{-k}}.\] For this fixed smooth metric, \(a(s)\to\omega_4\) as \(s\downarrow0\), where \(\omega_4\) is the volume of the Euclidean unit ball in \(\mathbb R^4\). Letting \(k\to\infty\) gives \(a(r)\ge c^2\). No uniform scale for the Euclidean asymptotic is required. ◻ Proposition 9. For all sufficiently large \(j\), the flow \(g_j(t)\) exists on \([0,T]\). There is a uniform constant \(A\) such that \[ \max_{M_j}|\mathop{\mathrm{Rm}}_{g_j(t)}|\le\frac At, \qquad 0<t<T. \tag{24}\] Proof. Fix \(A>2\) with \(A/T\ge1\) and \(A/T\ge4\max_{[0,T]}K\). Suppose that \(t\max_Mu(t)\) first reaches \(A\) at an existing time \(t<T\). Set \(Q=\max_Mu(t)=A/t\). On \([t-Q^{-1},t]\) the curvature is at most \(2Q\), because \(t-Q^{-1}\ge t/2\) and this is a first crossing. Parabolic rescaling gives a smooth closed flow on \([0,1]\) whose curvature norm is at most \(2\). Shi’s first derivative estimate (Shi 1989) gives \[|\nabla\mathop{\mathrm{Rm}}|(t)\le CQ^{3/2}.\] This estimate requires no initial derivative or injectivity-radius bound; one may use the local formulation in (Morgan and Tian 2007, Theorem 3.27) with the entire closed manifold as its domain. Let \(x\) realize the curvature maximum. Choose a fixed \(c\in(0,1]\) small enough that on \(B_{g_j(t)}(x,cQ^{-1/2})\) the derivative estimate implies \(u\ge3Q/4\). Since \(Q\ge4K(t)\), we have \(w\ge Q/2\) there. Also \(Q\ge1\), so this radius is at most \(1\). Lemma 8 now gives \[\int_Mw(t)^2\,d\mu\ge\frac{Q^2}{4}\,v_*c^4Q^{-2} =\frac{v_*c^4}{4}.\] This contradicts the uniform convergence of the tail norm to zero in Proposition 7. Thus, for large \(j\), no crossing occurs, and (24) holds throughout existing times below \(T\). If \(\mathcal T_j\le T\), the same bound controls curvature on \([\mathcal T_j/2,\mathcal T_j)\). Curvature is already bounded on the earlier compact interval by smoothness of the individual flow. The curvature continuation criterion contradicts maximality. Hence \(\mathcal T_j>T\). ◻ The Einstein and sectional-curvature defectsThe common interval now permits compactness at positive times. To identify the eventual limit, we must retain information from time zero: the Einstein error tends to zero, and the integral negative sectional curvature remains of order \(t\). Proposition 10. Set \(\lambda(t)=3/(1-6t)\). Along the flows above, \[ \sup_{0\le t<T} \left\lVert \mathop{\mathrm{Ric}}_{g_j(t)}-\lambda(t)g_j(t)\right\rVert_{L^2(g_j(t))} \longrightarrow0. \tag{25}\] Proof. Write \(L=\mathop{\mathrm{Ric}}-\lambda g\). In moving orthonormal frames, define the curvature action on a symmetric tensor \(B\) by \[(\mathop{\mathrm{Rm}}(B))_{ij}=\sum_{k,l}R_{ikjl}B_{kl}; \qquad \mathop{\mathrm{Rm}}(g)=\mathop{\mathrm{Ric}}.\] The Ricci evolution in these frames is \[\partial_t\mathop{\mathrm{Ric}}=\Delta\mathop{\mathrm{Ric}}+2\mathop{\mathrm{Rm}}(\mathop{\mathrm{Ric}}).\] The frame motion cancels the \(-2\mathop{\mathrm{Ric}}^2\) term in the fixed-frame formula; the components of \(g\) in the moving frames are constant. Since \(\lambda'=2\lambda^2\), subtraction yields \[ \partial_tL=\Delta L+2\mathop{\mathrm{Rm}}(L)+2\lambda L. \tag{26}\] Using \(R\ge0\), \(|\mathop{\mathrm{Rm}}|\le K+w\), and boundedness of \(K\) and \(\lambda\) on \([0,T]\), we obtain \[\frac d{dt}\int_M|L|^2\,d\mu \le-2\int_M|\nabla L|^2\,d\mu +C\int_M(1+w)|L|^2\,d\mu.\] By Hölder, (19), and Kato’s inequality, \[\int_Mw|L|^2\,d\mu \le\left\lVert w\right\rVert_2\left\lVert |L|\right\rVert_4^2 \le S_2\left\lVert w\right\rVert_2\int_M(|\nabla L|^2+|L|^2)\,d\mu.\] The coefficient tends to zero uniformly in time, by (20). Absorb the gradient term for large \(j\) and apply Gronwall’s inequality. The initial norm tends to zero by (6), proving (25). ◻ We next measure failure of nonnegative sectional curvature by a scalar function of the curvature tensor. On the Euclidean space of algebraic curvature tensors on \(\mathbb R^4\), define \[ F(B)=\max\left\{0, \sup_{\substack{|X|=|Y|=1\\\langle X,Y\rangle=0}} -B(X,Y,X,Y)\right\}. \tag{27}\] Thus \(F(\mathop{\mathrm{Rm}})\) is the magnitude of the most negative sectional curvature, or zero if all sectional curvatures are nonnegative. Proposition 11. There is a uniform constant \(C\) such that \[ \int_{M_j}F(\mathop{\mathrm{Rm}}_{g_j(t)})\,d\mu_{g_j(t)}\le Ct, \qquad 0\le t<T. \tag{28}\] Proof. The function \(F\) is nonnegative, convex, orthogonally invariant, and Lipschitz: it is a supremum of a bounded family of linear functionals and the zero function. Let \(F_\delta\) be its convolution with a nonnegative smooth radial mollifier supported in a ball of radius \(\delta\) in the algebraic-curvature tensor space. Then \(F_\delta\) is smooth, nonnegative, convex, and orthogonally invariant, with \[|DF_\delta|\le C,\qquad |F_\delta-F|\le C\delta.\] Orthogonal invariance makes this construction independent of the orthonormal frame and compatible with the metric connection. Apply the chain rule to (16) in moving orthonormal frames. Convexity gives a favorable Hessian term: \[\begin{align*} (\partial_t-\Delta)F_\delta(\mathop{\mathrm{Rm}}) &=DF_\delta(\mathop{\mathrm{Rm}})[\mathcal Q(\mathop{\mathrm{Rm}})] -\sum_aD^2F_\delta(\mathop{\mathrm{Rm}}) [\nabla_a\mathop{\mathrm{Rm}},\nabla_a\mathop{\mathrm{Rm}}]\\ &\le C|\mathop{\mathrm{Rm}}|^2. \end{align*}\] After integration, the volume variation contributes \(-\int R F_\delta(\mathop{\mathrm{Rm}})\,d\mu\le0\). Closedness and (21) therefore imply \[\int_MF_\delta(\mathop{\mathrm{Rm}}(t))\,d\mu \le\int_MF_\delta(\mathop{\mathrm{Rm}}(0))\,d\mu_{g_j}+Ct \le C\delta V_0+Ct.\] The last step uses \(F(\mathop{\mathrm{Rm}}(0))=0\). Letting \(\delta\downarrow0\) proves (28). ◻ The compact Einstein limit and the gapWe first extract the geometric conclusion that does not use Classification Assumption 1. Proposition 12. For any sequence (1), normalize the metrics as in Proposition 3 and discard finitely many terms. There are a subsequence, a connected simply connected closed smooth four-manifold \(N\), a metric \(h\) on \(N\), and diffeomorphisms \(\Phi_j:N\to M_j\) such that \[\mathop{\mathrm{Ric}}_h=3h,\qquad \sec_h\ge0,\] and \[ \Phi_j^*g_j(t)\longrightarrow (1-6t)h \tag{29}\] smoothly on \(N\times I\) for every compact interval \(I\subset(0,T)\). Proof. Fix \(t_*=T/2\). By (24), the flows have uniform global curvature bounds on every compact time interval in \((0,T)\). At \(t_*\), the local volume lower bound (23) and the total volume upper bound \(V_0\) also give a diameter bound. Indeed, points spaced a fixed distance apart along a minimizing geodesic have pairwise disjoint balls of a fixed smaller radius; each such ball has uniformly positive volume. Their number is bounded by \(V_0\). Write the resulting bound as \(\mathop{\mathrm{diam}}(M_j,g_j(t_*))\le D_*\). Absolute sectional curvature bounds and the ball-volume lower bound give a uniform positive injectivity radius at \(t_*\), by the injectivity radius estimate of Cheeger–Gromov–Taylor (Cheeger et al. 1982, Theorem 4.7). Hamilton’s compactness theorem (Hamilton 1995) now gives, after passing to a subsequence, smooth pointed convergence to a connected Ricci flow \((N,h(t))\) on \((0,T)\), with \(h(t_*)\) complete. One can apply the theorem on nested compact time intervals containing \(t_*\) and take a diagonal subsequence. The curvature bounds used here are global in space; hence the complete limit version of compactness applies, without the completeness issue that can arise from purely local bounds (Topping 2014, Theorem 1.1 and subsequent discussion). The convergence maps are time-independent embeddings defined on an exhaustion of \(N\). We explain why the limit is compact and the embeddings are eventually diffeomorphisms. If the complete connected slice \((N,h(t_*))\) were noncompact, it would contain a compact closed ball \(\overline B(p,L)\) with a point \(q\) at distance \(L\), for any \(L>0\). Choose \(L>2D_*\). For large \(j\), a convergence embedding is defined on a neighborhood of this ball, and its pulled-back metric bounds lengths on the ball from below by half their \(h(t_*)\) values. Any curve in \(M_j\) joining the images of \(p\) and \(q\) must have length at least \(L/2\): if it leaves the image of the ball, its initial portion must first reach that image’s boundary and already has this length. If it stays inside, the same bound follows by pullback. This contradicts the diameter bound. Thus \(N\) is compact. Eventually the convergence embeddings are defined on all of \(N\); their images are open, by the inverse function theorem, and closed, by compactness. Since \(M_j\) is connected, each is a diffeomorphism. It follows also that \(N\) is simply connected. Smooth convergence on this compact manifold and (25) give \[\mathop{\mathrm{Ric}}_{h(t)}=\lambda(t)h(t),\qquad 0<t<T.\] The Ricci-flow equation therefore integrates to \[h(t)=(1-6t)h,\qquad h=\frac{1}{1-6t_*}h(t_*),\qquad \mathop{\mathrm{Ric}}_h=3h.\] For each fixed \(t>0\), (28) also passes to the limit. The function \(F\) is positively homogeneous of degree one. Under metric scaling by a positive constant \(c\), its value on curvature scales by \(c^{-1}\) and the four-dimensional volume form scales by \(c^2\). Consequently \[ (1-6t)\int_NF(\mathop{\mathrm{Rm}}_h)\,d\mu_h\le Ct, \qquad 0<t<T. \tag{30}\] Let \(t\downarrow0\). The nonnegative continuous integrand must vanish, so \(\sec_h\ge0\). This proves the proposition. ◻ Proof of Theorem 2. If no universal positive gap existed, then for every positive integer \(j\) there would be a connected simply connected closed four-manifold \((M_j,g_j)\) with \(\sec_{g_j}\ge0\) and \(\mathcal D(g_j)<1/j\), whose smooth manifold is outside the three stated diffeomorphism types. Proposition 12 gives a subsequence diffeomorphic to a closed simply connected \((N,h)\) with \(\mathop{\mathrm{Ric}}_h=3h\) and \(\sec_h\ge0\). Classification Assumption 1 applies exactly to this metric. Since \(N\) is already simply connected, it is one of the three model manifolds, a contradiction. ◻ Remark 13. Proposition 12 is independent of the classification input. It asserts smooth convergence only after positive-time regularization. In particular, the recovery of \(\sec_h\ge0\) in (30) requires neither convergence of the original metrics at time zero nor preservation of nonnegative sectional curvature by Ricci flow.
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