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  "body": "\n## Statement\n\nLet $M$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] without boundary and with negative [[sectional curvature|1X42JAA349]], with volume measure $dv$ and total volume $V$.\nLet $(\\varphi_j)$ be any sequence of $L^2$-normalised eigenfunctions of the [[Laplace–Beltrami operator|F4NWZNTRMD]] with eigenvalues $\\lambda_j \\to \\infty$.\nThe conjecture says that for every continuous function $a$ on $M$\n$$\\lim_{j \\to \\infty} \\int_M a \\, \\abs{\\varphi_j}^2 \\, dv = \\frac{1}{V} \\int_M a \\, dv.$$\nIn its full form it says that the microlocal lifts of the measures $\\abs{\\varphi_j}^2 \\, dv$ converge to the normalised Liouville measure on the unit cotangent bundle.\n\n## Remarks\n\n[[Quantum ergodicity|D9W79NKYZJ]] gives this convergence along a subsequence of density one; the conjecture asks that there be no exceptional subsequence at all.\nThe conjecture is open in general.\nLindenstrauss proved the analogue for compact arithmetic hyperbolic surfaces and joint eigenfunctions of the Laplacian and the Hecke operators.\nTogether with work of Soundararajan that rules out escape of mass, this also covers the modular surface.\nAnantharaman proved that on compact manifolds with Anosov geodesic flow every quantum limit has positive entropy, so eigenfunctions cannot concentrate entirely on a closed geodesic.\nFor some ergodic billiards the analogue fails: Hassell proved that for almost every stadium-shaped billiard there are exceptional sequences of eigenfunctions.\n\n## Explanations\n\n### L1 human\n\nOn a negatively curved surface, a freely moving particle explores every region evenly.\nThe conjecture says that every high-frequency standing wave on such a surface also spreads its mass evenly, with no exceptions at all.\nQuantum ergodicity already proves this for almost all waves; the conjecture removes the word \"almost\".\n",
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      "first_stated": "1994",
      "posed_by": "Zeév Rudnick and Peter Sarnak"
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