Decreasing Weyl's energy by connected sums with locally conformally flat manifolds

Decreasing Weyl’s Energy via Connected Sums with Locally Conformally Flat Manifolds

Introduction and Context

The paper investigates the behavior of the Weyl functional,

$\mathcal{W}(g) = \int_M |W_g|^2\, dV_g,$

on four-dimensional manifolds under the operation of connected sum, focusing on the interplay between Bach-flat metrics and locally conformally flat (LCF) geometries. The Weyl functional is conformally invariant in dimension four and is extremized at Bach-flat metrics. The main goal is to understand if, and under what circumstances, the Weyl energy of a connected sum $Y = M \# Z$ can be made less than that of one or both original manifolds, with $M$ Bach-flat and $Z$ LCF.

Crucial technical distinctions are drawn between manifolds that are self-dual or anti-self-dual, and the dichotomy between positive and non-positive Yamabe classes. The construction leverages fine analysis of the structure of Weyl energy under high-precision gluing, engaging the topology of $Z$ and both (anti-)self-dual components of $W_M$.

Main Results and Techniques

The centerpiece is the following: if $g_M$ is Bach-flat, neither self-dual nor anti-self-dual, and $g_Z$ is LCF of positive Yamabe class but not globally conformal to the round sphere, then there exists a metric $g_Y$ on $Y = M \# Z$ with

$Y = M \# Z$0

This strict inequality fails only in the trivial case where $Y = M \# Z$1 is the standard sphere.

Significant secondary results include variants that accommodate orbifold singularities, leveraging recently developed classifications of LCF orbifolds with positive scalar curvature. The approach extends to a class of connected sums involving orbifold points with prescribed isotropy.

The proof strategy synthesizes the following elements:

• Blow-up/Blow-down and Conformal Inversion: A conformal blow-up is performed on $Y = M \# Z$2 at a point $Y = M \# Z$3, yielding an ALE manifold whose asymptotic region is matched to a punctured neighborhood in $Y = M \# Z$4, itself rescaled to fit the neck produced by the blow-up.
• Green Kernel Corrections: The matching involves solving for a “singular correction tensor” to $Y = M \# Z$5, with a prescribed pole at the connecting point and leading singular part matching the principal part of the conformal inversion of $Y = M \# Z$6. The analysis is particularly subtle when $Y = M \# Z$7 is a non-trivial quotient or connected sum of standard LCF geometries.
• Gauge Corrections and the Linearized Bach Operator: Critical use is made of the structure of the kernel of the linearized Bach operator, especially Lie derivatives of the metric, to adjust the constructed tensors for suitable asymptotic and equivariance properties. This technical machinery ensures correct matching and allows for controlling higher-order terms in the expansion of the Weyl energy.
• Quantitative Expansion and Interaction Analysis: The energy computation yields interaction terms involving contractions like $Y = M \# Z$8, where $Y = M \# Z$9 arises from the aforementioned correction. The sign and size are studied in detail, showing that unless $M$0 is self-dual or anti-self-dual, the interaction is strictly negative for suitable choices.
• Conformal and Topological Invariance, and Degeneration Analysis: The result is robust under changes of orientation and connected sum points, and also accommodates bubbling and orbifold limits by virtue of the Weyl energy expansion structure.

Numerical and Qualitative Implications

A key quantitative implication is:

$M$1

unless $M$2 is the standard sphere, or $M$3 is self-dual/anti-self-dual (in which case, conjecturally, a “neck” minimizer is degenerate and equality is attained). The explicit leading term in the energy expansion is proportional to the pairing between the Weyl tensors, with a coefficient encapsulating the underlying ADM-like mass of the LCF component’s blow-up.

In orbifold extensions, the energy decrease is preserved except when $M$4 is an orbifold football with no nontrivial LCF structure other than the football itself—again, a class covered by rigidity results.

Theoretical and Practical Implications

Theoretical

• Obstructions to Topology-Changing Degeneration:

The energy comparison constrains possible bubbling phenomena in blow-up analysis for sequences minimizing the Weyl energy, imposing topological rigidity in the formation of singularities for critical sequences.

• Relation to the Singer Conjecture and Classification:

The case where both summands are self-dual or anti-self-dual is intimately connected with the Singer conjecture. The results clarify and isolate the “singular” case and reinforce the centrality of self-duality in the energy landscape for four-manifolds.

• Extending Analytic Gluing to Orbifolds:

By adapting the construction to orbifold settings—in particular, allowing for singularities at the bubble points—the methods inform the analytic theory of Bach-flat metrics with isolated singularities, ALE or ALE-like ends.

• Comparison with Willmore Surface Theory:

The gluing and energy estimate strategy runs in close parallel with known methods for Willmore minimization in higher-genus surfaces, confirming an “energy-deficit” picture for connected sum formation in curvature functionals of critical order.

Practical

• Construction of New Bach-flat Metrics:

The result enables the construction of new metrics with strictly lower Weyl energy on a rich class of connected sums, potentially useful for exploring moduli and classification problems in four-dimensional conformal geometry.

• Providing Sharp Energy Identities for Analysis:

The leading order term and error estimates are sharp and support arguments to preclude certain degenerations in the compactness analysis of minimizing sequences for conformally invariant functionals.

• Boundary of Conformal Invariance Effects:

The orbifold and LCF quotient extension demonstrates how conformal invariance interacts with topological complexity—sharpening the landscape for permissible geometric structures when minimizing Weyl energy among Bach-flat metrics.

Speculation on Future Developments

• Complete Compactness/Convergence Theory for Minimizers:

Given these results, further progress toward a full compactness theorem for sequences minimizing the Weyl functional (under non-collapsing hypotheses) in four-manifolds appears accessible, potentially excluding all but self-dual/anti-self-dual degeneration scenarios.

• Extensions to Higher Dimensions or Other Conformally Invariant Functionals:

The basic analytic and gluing techniques may generalize, after technical modifications, to dimensions $M$5 and other critical (e.g., Paneitz) invariants, though conformal invariance fails in higher even dimensions.

• Refined Analysis of Singularities and Higher-Order Obstructions:

In orbifold settings, deeper investigation into higher-order vanishing of Weyl on the “limit” spaces could yield new energy identities and further refine the landscape of possible degeneration/compactness obstructions.

• Interplay with Geometric Flows:

The Bach flow’s analytic tractability has been enhanced by these structural results. Coupling these energy inequalities with flow-analytic control techniques (à la Ricci and Yamabe flow) could yield new smoothing or long-time existence results for curvature flows targeting Bach-flat or critical metrics.

Conclusion

The paper provides a rigorous quantitative reduction of the Weyl functional for connected sums with LCF four-manifolds, excluding the trivial case of the round sphere. This result clarifies possible energy-lowering topological surgeries available for constructing Bach-flat metrics and supplies strong analytical tools for understanding minimizing sequences' behavior in conformal geometry. The analysis succeeds in bridging topological, analytic, and conformal geometric perspectives, with additional reach into orbifold and degeneration settings. The implications for four-manifold geometry and the analytic theory of critical curvature functionals are both wide and profound.