Qiongling Li’s pointwise energy-monotonicity conjecture

Let $X$ be a compact Riemann surface and let $(E,\Phi)$ be a stable $\mathrm{SL}(n,\mathbb{C})$ Higgs bundle on $X$ . For each $r>0$ , let $e_r(x)$ denote the energy density at $x\in X$ associated with the Higgs bundle $(E,r\Phi)$ and its harmonic metric. The conjecture asserts that for every $x\in X$ and all $0<r\leq s$ , $e_r(x)\leq e_s(x)$ ; equivalently, the energy density is pointwise nondecreasing along the positive-real part of the $\mathbb{C}^*$ -orbit of $(E,\Phi)$ .

References

Primary source

arXiv

Additional references

Pointwise energy monotonicity for stable Higgs bundles — arXiv — Tianzhi Hu

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new preprint claims the conjecture is true in rank two but false in higher ranks, leaving only verification of the reported results.

The conjecture concerns whether energy density is pointwise nondecreasing along the $C^*$ flow for stable Higgs bundles. The new report identifies rank two as the sharp setting in which this behavior is claimed to hold.

September 28, 2026 rank-two proof and higher-rank counterexamples

Tianzhi Hu’s preprint Pointwise energy monotonicity for stable Higgs bundles reports pointwise nondecrease in rank two and stable higher-rank examples violating monotonicity. Thus it claims both a proof of the rank-two case and a refutation of the unrestricted higher-rank conjecture; these claims remain unverified here.

Current status (as of September 2026): the unrestricted conjecture is claimed false in higher rank, while the rank-two case is claimed proved; neither claim has independent verification in the retrieved record.

Sources

No solutions have been posted yet.