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Filip Broćić

Publications and source records attributed to Filip Broćić.

10 recordsLinked to original sources
arXiv2026

Lengths of Reeb chords and Viterbo restriction

Let $Λ$ be a Legendrian in the contact boundary of a Liouville domain $Ω$. We explain how the non-existence of Reeb chords with endpoints on $Λ$ of length up to $a$ enables one to embed $D_εT^{*}Λ\times D(a)$ into $Ω$ in an exact way. As in earlier work of Zhengyi Zhou, we use the Viterbo restriction map to deduce a contradiction in certain cases. In particular, we show that if $M$ admits a submersion from a product of spheres (e.g., the $n$-torus), then all compact Legendrians $Λ\subset ST^{*}M$ admit a Reeb chord for every choice of contact form $ST^{*}M$. The obstruction we use in this case is based on the idea of inverting the degree-$n$ classes in cohomology, and is similar to the notion of string point invertibility introduced by Egor Shelukhin.

math.SG↗
arXiv2025

A note on the wild symplectic ellipsoids

We show that the symplectic $2$-product of $n$ two-dimensional star-shaped domains has an interior symplectomorphic to that of a symplectic ellipsoid. Adapting this construction, given $0<α\leq 1$, we obtain that every open subset of $\mathbb{R}^{2n}$ with a smooth boundary is symplectomorphic to an open set whose boundary contains a set of Hausdorff dimension $2n-1+α$.

math.SG↗
arXiv2025

Wrapped Floer homology and subcritical handle attachment

In this expository article, we present the proof of the invariance of the wrapped Floer homology under the subcritical handle attachment. This is proved by Irie. Here, we fix a minor gap in the proof about the choice of a cofinal family of Hamiltonians. We adapt the arguments from Fauck's PhD thesis, who resolved the gap for the case of handle attachment in symplectic homology. The effect of the handle attachment on the symplectic homology was originally explored by Cieliebak.

math.SG↗
arXiv2025

Wrapped Floer homology and the circular restricted three-body problem

Using the wrapped Floer homology, we prove the existence of consecutive collisions at the primaries in the circular restricted three-body problem. We also prove the existence of a symmetric periodic orbit. These existence results are obtained for energy hypersurfaces slightly above the first critical value.

math.SG↗
arXiv2025

Parametric Gromov width of Liouville domains

The classical Gromov width measures the largest symplectic ball embeddable into a symplectic manifold; inspired by the symplectic camel problem, we generalize this to ask how large a symplectic ball can be embedded as a family over a parameter space $N$. Given a smooth map $f: N \to Ω$, where $Ω$ is a symplectic manifold, we define the \emph{parametric Gromov width} $\mathrm{Gr}(f,Ω)$ as the supremum of capacities $a>0$ for which there exists a family of balls, parametrized by $N$, of capacity $a$ whose centers trace out the map $f$. For Liouville domains $Ω$, we establish upper bounds on $\mathrm{Gr}(f,Ω)$ using the Floer cohomology persistence module associated to $Ω$. Specializing to fiberwise starshaped domains in the cotangent bundle $T^*M$, we derive computable bounds via filtered string topology. Specific examples of $Ω$ -- including disk cotangent bundles of thin ellipsoids, open books, and tori -- demonstrate our bounds, and reveal constraints on parameterized symplectic embeddings beyond the classical Gromov width.

math.SG↗
arXiv2024

A note on the capacities of Lagrangian $p$-sum

In this short note, we construct an explicit embedding of the rescaling of the $p$-sum $K\oplus_p K^{\circ}$ of the centrally symmetric convex domain $K$ and its polar $K^{\circ}$ to the product $K \times K^{\circ}$. The rescaling constant is sharp in some cases. Additionally, we comment on the strong Viterbo conjecture for $K\oplus_p K^{\circ}$.

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arXiv2024

The chord conjecture for conormal bundles

We prove Arnol'd's chord conjecture for all Legendrian submanifolds of cosphere bundles of closed manifolds isotopic to conormal bundles of closed submanifolds. Our method of proof involves an isomorphism between wrapped Floer cohomology and the homology of a path space with coefficients in a local system and a twisted version of the Hurewicz theorem.

math.SG↗
arXiv2023

Riemannian distance and symplectic embeddings in cotangent bundle

Given an open neighborhood $W$ of the zero section in the cotangent bundle of $N$ we define a distance-like function $ρ_W$ on $N$ using certain symplectic embeddings from the standard ball $B^{2n}(r)$ to $W$. We show that when $W$ is the unit disc-cotangent bundle of a Riemannian metric on $N$, $ρ_W$ recovers the metric. As an intermediate step, we give a new construction of the ball of capacity 4 to the product of Lagrangian discs $P_L := B^n(1)\times B^n(1)$, and we give a new proof of the strong Viterbo conjecture about normalized capacities for $P_L$. We also give bounds of the symplectic packing number of two balls in a unit disc-cotangent bundle relative to the zero section $N$.

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arXiv2023

Bordism classes of loops and Floer's equation in cotangent bundles

For each representative $\mathfrak{B}$ of a bordism class in the free loop space of a manifold, we associate a moduli space of finite length Floer cylinders in the cotangent bundle. The left end of the Floer cylinder is required to be a lift of one of the loops in $\mathfrak{B}$, and the right end is required to lie on the zero section. Under certain assumptions on the Hamiltonian functions, the length of the Floer cylinder is a smooth proper function, and evaluating the level sets at the right end produces a family of loops cobordant to $\mathfrak{B}$. The argument produces arbitrarily long Floer cylinders with certain properties. We apply this to prove an existence result for 1-periodic orbits of certain Hamiltonian systems in cotangent bundles, and also to estimate the relative Gromov width of starshaped domains in certain cotangent bundles. The moduli space is similar to moduli spaces considered by Abbondandolo-Schwarz and Abouzaid for Tonelli Hamiltonians. The Hamiltonians we consider are not Tonelli, but rather of ``contact-type'' in the symplectization end.

math.SG↗