We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma \subset \mathbb{R}^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3‑dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth ‘spacetime harmonic functions’ in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

For a harmonic map $u : M^3 \to S^1$ on a closed, oriented $3$-manifold, we establish the identity\[2 \pi \int_{\theta \in S^1} \chi (\Sigma _\theta) \geq \frac{1}{2} \int_{\theta \in S^1} \int_{\Sigma_\theta}\left( {\lvert du \rvert}^{-2} {\lvert \mathit{Hess} (u) \rvert}^2 + R_M \right)\]relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $\Sigma_\theta = u^{-1} {\lbrace \theta \rbrace}$. As our primary application, we extend the Kronheimer–Mrowka characterization of the Thurston norm on $H_2 (M; \mathbb{Z})$ in terms of ${\lVert R^{-}_M \rVert}_{L^2}$ and the harmonic norm to any closed $3$-manifold containing no nonseparating spheres. Additional corollaries include the Bray–Brendle–Neves rigidity theorem for the systolic inequality $(\min R_M) \mathit{sys}_2 (M) \leq 8\pi$, and the well-known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.

We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the cohomologies of primitive forms on manifolds with boundary. We further use these boundary conditions to define a relative version of the primitive cohomologies and to relate primitive cohomologies with Lefschetz maps on manifolds with boundary. As we show, these cohomologies of primitive forms can distinguish certain Kähler structures of Kähler manifolds with boundary.

We prove a sharp relative Clifford inequality for relatively special divisors on varieties fibered by curves. It generalizes the classical Clifford inequality about a single curve to a fibration of curves. It yields a geographical inequality for varieties Albanese-fibered by curves. We also apply it to deduce a slope inequality for some higher dimensional families of curves. It sheds light on the existence of a more general Cornalba–Harris–Xiao type inequality for families of curves.