Inhomogeneous Einstein metrics on complex projective spaces

Abstract: The only Einstein metrics currently known on complex projective spaces are homogeneous: the Fubini-Study metric, arising via the Hopf fibration; and, in odd complex dimensions, Ziller's metric, obtained as a canonical variation along the twistor fibration over the quaternionic projective space. In 1965, Berger proved that the Fubini-Study metric is the unique Kähler-Einstein metric on the complex projective space, and posed the question of whether other Einstein metrics exist. We construct the first inhomogeneous Einstein metrics on complex projective spaces of complex dimension $n$ for $3\leq n \leq 7$, answering Berger's question affirmatively for the complex even dimensional cases $n=4$ and $n=6$.

• The paper constructs the first inhomogeneous Einstein metrics on complex projective spaces for n=3,4,5,6,7 by reducing the problem to a four-dimensional cohomogeneity-one matching system and rigorously certifying a solution with interval arithmetic and Krawczyk’s theorem.
• The resulting SO(n+1)-invariant metrics are genuinely inhomogeneous, non-Hermitian, not nearly Kähler, and do not have non-negative sectional curvature, while their Yamabe energy is numerically 6.4–22.2% lower than that of the Fubini–Study metric.
• The paper answers Berger’s 1965 question about a second Einstein metric on CP^n, establishes complex dimension 3 as the minimal case, and leaves existence for n≥8 and classification of Hermitian–Einstein metrics as open problems.

The construction of new Einstein metrics on compact manifolds with high symmetry is a long-standing problem in Riemannian geometry. This paper by Cao-Labora and Rodríguez-Vázquez constructs the first inhomogeneous Einstein metrics on complex projective spaces $\mathbb{C}P^n$, for $n \in \{3,4,5,6,7\}$, answering affirmatively a question posed by Berger in 1965 concerning the existence of a second Einstein metric on $\mathbb{C}P^n$. Prior to this work, the only known Einstein metrics on $\mathbb{C}P^n$ were the homogeneous ones: the Fubini–Study metric and, in odd complex dimension, Ziller's metric obtained via a canonical variation along the twistor fibration over $\mathbb{H}P^n$ (2608.16880). The new metrics are $\mathrm{SO}_{n+1}$-invariant, of cohomogeneity one, non-Hermitian, and not nearly Kähler; their existence is established by a computer-assisted argument based on interval arithmetic and Krawczyk's fixed-point theorem.

Geometric setup and the cohomogeneity-one action

The construction exploits the standard $\mathrm{SO}_{n+1}$-action on $\mathbb{C}P^n = S^{2n+1}/S^1$, which descends to a cohomogeneity-one action whose orbit space is an interval. The two singular orbits are a totally geodesic Lagrangian $\mathbb{R}P^n$ (the fixed-point set of complex conjugation) and the complex quadric $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$; principal orbits are diffeomorphic to the unit tangent bundle $n \in \{3,4,5,6,7\}$0. The group diagram is $n \in \{3,4,5,6,7\}$1.

Any $n \in \{3,4,5,6,7\}$2-invariant metric on the regular part takes the diagonal form

$n \in \{3,4,5,6,7\}$3

where $n \in \{3,4,5,6,7\}$4 is the isotropy decomposition with $n \in \{3,4,5,6,7\}$5 and $n \in \{3,4,5,6,7\}$6 the standard $n \in \{3,4,5,6,7\}$7-module. A key point, established via Dammermann's diagonalization theorem, is that for $n \in \{3,4,5,6,7\}$8 every $n \in \{3,4,5,6,7\}$9-invariant Einstein metric on $\mathbb{C}P^n$0 admits this diagonal form, so the ansatz is exhaustive in that range. Smoothness conditions at the two singular orbits, derived from the general theory of Verdiani–Ziller, impose that the local solution space at each endpoint is two-dimensional — a structural feature that distinguishes this problem from the classical two-function $\mathbb{C}P^n$1 ansatz on spheres, where the matching problem is only two-dimensional.

The Einstein equations $\mathbb{C}P^n$2 reduce to a system of four ODEs (equations (E1)–(E4)) for $\mathbb{C}P^n$3, which are dependent via the Eschenburg–Wang first integral relating $\mathbb{C}P^n$4, $\mathbb{C}P^n$5, and $\mathbb{C}P^n$6 to $\mathbb{C}P^n$7. The Fubini–Study metric, with $\mathbb{C}P^n$8, $\mathbb{C}P^n$9, $\mathbb{C}P^n$0 and $\mathbb{C}P^n$1, is a solution.

The $\mathbb{C}P^n$2-reparametrization and the four-dimensional matching problem

The authors reparametrize the autonomous six-dimensional ODE system using the mean curvature $\mathbb{C}P^n$3 of the principal orbits. By the Riccati equation and Cauchy–Schwarz, $\mathbb{C}P^n$4, so $\mathbb{C}P^n$5 is a strictly monotone function of $\mathbb{C}P^n$6 mapping $\mathbb{C}P^n$7 onto $\mathbb{C}P^n$8, with asymptotics $\mathbb{C}P^n$9 near the $\mathbb{H}P^n$0 orbit and $\mathbb{H}P^n$1 near the quadric. In the scale-invariant variables $\mathbb{H}P^n$2, $\mathbb{H}P^n$3 and logarithmic derivatives $\mathbb{H}P^n$4, the system becomes a four-dimensional non-autonomous ODE in $\mathbb{H}P^n$5. A converse proposition reconstructs $\mathbb{H}P^n$6 from any solution of this system satisfying sign conditions on $\mathbb{H}P^n$7 and on $\mathbb{H}P^n$8, so solving the $\mathbb{H}P^n$9-system is fully equivalent to solving the Einstein equations.

Each singular orbit admits a two-parameter family of smooth local solutions, encoded in convergent series in $\mathrm{SO}_{n+1}$0: near $\mathrm{SO}_{n+1}$1 the free parameters are $\mathrm{SO}_{n+1}$2; near $\mathrm{SO}_{n+1}$3 they are $\mathrm{SO}_{n+1}$4 in the symmetric variables adapted to the quadric. Gluing these two families into a global solution requires matching four quantities at the minimal orbit ($\mathrm{SO}_{n+1}$5), yielding a four-dimensional matching function $\mathrm{SO}_{n+1}$6. The existence of a smooth Einstein metric is thus equivalent to $\mathrm{SO}_{n+1}$7 having a zero in a suitable region.

Series expansions and computer-assisted proof

Local solutions at both endpoints are constructed by explicit power series in $\mathrm{SO}_{n+1}$8, with recurrences determined by the free parameters. Convergence is proved by propagating Catalan-number bounds on the Taylor coefficients, yielding uniform convergence on half-lines $\mathrm{SO}_{n+1}$9 and $\mathrm{SO}_{n+1}$0 together with explicit remainder estimates and analytic dependence on the parameters — the latter needed for rigorous derivative bounds via Cauchy estimates.

The zero of $\mathrm{SO}_{n+1}$1 is then certified with Krawczyk's fixed-point theorem: if the Krawczyk box $\mathrm{SO}_{n+1}$2 is contained in the interior of a small box $\mathrm{SO}_{n+1}$3 around an approximate root $\mathrm{SO}_{n+1}$4, then $\mathrm{SO}_{n+1}$5 has a genuine zero in $\mathrm{SO}_{n+1}$6. The interval-arithmetic implementation combines the series enclosures with the CAPD library for rigorous ODE continuation to $\mathrm{SO}_{n+1}$7. This is the only computer-assisted step in the paper, and it is structurally different from prior CAP constructions of Einstein metrics on spheres: instead of a Schauder fixed-point argument on an infinite-dimensional Banach space, the computer certifies a zero of a finite-dimensional (four-dimensional) matching map. The authors note this technique should extend to ansätze depending on more functions.

Properties of the new metrics

The resulting metrics have several notable features, each established within the corresponding section of the paper:

• Inhomogeneity: all homogeneous Einstein metrics on $\mathrm{SO}_{n+1}$8 (Fubini–Study and, in odd dimension, Ziller's) are geodesic orbit spaces, so their Jacobi operator has constant spectrum along every geodesic. For the new metrics, $\mathrm{SO}_{n+1}$9 takes both positive and negative values on $\mathbb{C}P^n = S^{2n+1}/S^1$0, so the Jacobi operator along the normal geodesic has non-constant spectrum, proving inhomogeneity. This argument simultaneously shows the metrics do not have non-negative sectional curvature.
• Non-Hermiticity: an $\mathbb{C}P^n = S^{2n+1}/S^1$1-invariant metric that is Hermitian with respect to the standard complex structure must satisfy $\mathbb{C}P^n = S^{2n+1}/S^1$2 along the geodesic, forcing $\mathbb{C}P^n = S^{2n+1}/S^1$3; the constructed metrics satisfy $\mathbb{C}P^n = S^{2n+1}/S^1$4 since the certified $\mathbb{C}P^n = S^{2n+1}/S^1$5. Hence they are not $\mathbb{C}P^n = S^{2n+1}/S^1$6-Hermitian and, a fortiori, not conformally Kähler.
• Not nearly Kähler: for $\mathbb{C}P^n = S^{2n+1}/S^1$7 this follows from Nagy's classification of strictly nearly Kähler manifolds combined with the known cases of the LeBrun–Salamon conjecture in real dimensions 8, 12, 16; for $\mathbb{C}P^n = S^{2n+1}/S^1$8 it follows by comparing the singular orbit structure with the unique $\mathbb{C}P^n = S^{2n+1}/S^1$9-preserving cohomogeneity-one action.
• Yamabe energy: numerically, the new metrics have smaller Yamabe energy than Fubini–Study, with ratios $\mathbb{R}P^n$0 increasing from $\mathbb{R}P^n$1 at $\mathbb{R}P^n$2 to $\mathbb{R}P^n$3 at $\mathbb{R}P^n$4, while the corresponding ratios for Ziller's metric decrease with dimension — a monotonicity phenomenon the authors highlight without explanation.

The Main Theorem also settles the minimal dimension: combining the result with Broder's classification of cohomogeneity-one Einstein metrics on closed 4-manifolds, $\mathbb{R}P^n$5 (real dimension 6) is the smallest complex dimension admitting an inhomogeneous cohomogeneity-one Einstein metric on $\mathbb{R}P^n$6.

Limitations and open questions

The paper is explicit about the scope of its result. The existence proof is confined to $\mathbb{R}P^n$7; numerical exploration suggests no $\mathbb{R}P^n$8-invariant solutions exist for $\mathbb{R}P^n$9, but this is not proved. The cases $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$0 and $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$1 are topologically distinguished — the principal orbits $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$2 are diffeomorphic to $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$3 exactly in these dimensions — and $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$4 is analytically special because the linear system in the recurrence degenerates ($$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$5), consistent with Broder's classification. The authors also flag an error in a related preprint (de Araujo–Grajales–Grama) claiming no totally geodesic singular orbits are possible, which is contradicted by both Fubini–Study and the new metrics. Two questions remain open: whether $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$6 admits inhomogeneous Einstein metrics for $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$7 under any symmetry, and whether every Hermitian–Einstein metric on $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$8 with $$Q_{n-1} \cong \mathrm{SO}_{n+1}/(\mathrm{SO}_2 \times \mathrm{SO}_{n-1})$$9 is homogeneous.

Conclusion

This paper resolves a sixty-year-old question by producing inhomogeneous Einstein metrics on $n \in \{3,4,5,6,7\}$00 for $n \in \{3,4,5,6,7\}$01 via a cohomogeneity-one $n \in \{3,4,5,6,7\}$02-invariant ansatz. Methodologically, its contribution is a finite-dimensional matching strategy — convergent series with Catalan-type coefficient bounds at the singular orbits, combined with interval-arithmetic ODE continuation and Krawczyk certification — that avoids infinite-dimensional fixed-point arguments and is designed to generalize to ansätze with more than three functions. The metrics themselves are non-Hermitian, not nearly Kähler, lack non-negative sectional curvature, and carry lower Yamabe energy than the homogeneous examples, and their existence in even complex dimension $n \in \{3,4,5,6,7\}$03 had no prior analogue.