Reine Angew Research Articles

Tropical Mirror Symmetry in Dimension One

We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [B\"ohm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.

Root finding algorithms and persistence of Jordan centrality in growing random trees

We consider models of growing random trees {Tf(n):n≥1} with model dynamics driven by an attachment function f:Z+→R+. At each stage a new vertex enters the system and connects to a vertex v in the current tree with probability proportional to f(degree(v)). The main goal of this study is to understand the performance of root finding algorithms. A large body of work (e.g., Random Structures Algorithms 50 (2017) 158–172; IEEE Trans. Netw. Sci. Eng. 4 (2017) 1–12; Random Structures Algorithms 52 (2018) 136–157) has emerged in the last few years in using techniques based on the Jordan centrality measure (J. Reine Angew. Math. 70 (1869) 185–190) and its variants to develop root finding algorithms. Given an unlabeled unrooted tree, one computes the Jordan centrality for each vertex in the tree and for a fixed budget K outputs the optimal K vertices (as measured by Jordan centrality). Under general conditions on the attachment function f, we derive necessary and sufficient bounds on the budget K(ε) in order to recover the root with probability at least 1−ε. For canonical examples such as linear preferential attachment and uniform attachment, these general results give matching upper and lower bounds for the budget. We also prove persistence of the optimal K Jordan centers for any K, that is, the existence of an almost surely finite random time n∗ such that for n≥n∗ the identity of the K-optimal Jordan centers in {Tf(n):n≥n∗} does not change, thus describing robustness properties of this measure. Key technical ingredients in the proofs of independent interest include sufficient conditions for the existence of exponential moments for limits of (appropriately normalized) continuous time branching processes within which the models {Tf(n):n≥1} can be embedded, as well as rates of convergence results to these limits.

Pairing and duality of algebraic quantum groupoids

Algebraic quantum groupoids have been developed by two of the authors of this note (AVD and SHW) in a series of papers [A. Van Daele and S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, Banach Center Publ. 98 (2012) 367–415; A. Van Daele and S. Wang, Weak multiplier Hopf algebras I. The main theory, J. Reine Angew. Math. 705 (2015) 155–209; A. Van Daele and S. Wang, Weak multiplier Hopf algebras II. The source and target algebras, preprint (2014), arXiv: 1403.7906v2 [math.RA]; A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], see also [A. Van Daele, Algebraic quantum groupoids. An example, preprint (2017), arXiv: 1702.04903 [math.RA]]. By an algebraic quantum groupoid, we understand a regular weak multiplier Hopf algebra with enough integrals. Regular multiplier Hopf algebroids are obtained also by two authors of this note (TT and AVD) in [T. Timmermann and A. Van Daele, Regular multiplier Hopf algebroids. Basic theory and examples, Commun. Algebra 46 (2017) 1926–1958]. Integral theory and duality for those have been studied by one author here (TT) in [T. Timmermann, Integration on algebraic quantum groupoids, Int. J. Math. 27 (2016) 1650014, arXiv: 1507.00660 [QA]; T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]. In these papers, the term algebraic quantum groupoid is used for a regular multiplier Hopf algebroid with a single faithful integral. Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. In the paper, Weak multiplier Hopf algebras III. Integrals and duality [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]], one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper, On duality of algebraic quantum groupoids [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]], a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids (see [T. Timmermann, On duality of algebraic quantum groupoids, Adv. Math. 309 (2017) 692–746, arXiv: 1605.06384 [math.QA]]). That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way followed in [A. Van Daele and S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint (2017), arXiv: 1701.04951 [math.RA]]. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids as studied in [T. Timmermann and A. Van Daele, Multiplier Hopf algebroids arising from weak multiplier Hopf algebras, Banach Center Publ. 106 (2015) 73–110]. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals. We are convinced that the material we present in this paper will provide a deeper understanding of the duality of algebraic quantum groupoids, both within the framework of weak multiplier Hopf algebras, as well as more generally for multiplier Hopf algebroids. Finally, we feel it is also appropriate to include some historical comments on the development of these duality theories.

Area minimizing surfaces in homotopy classes in metric spaces

We introduce and study a notion of relative 1 1 -homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1 1 -homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1 1 -homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1 1 -homotopy class. Our theorems generalize and strengthen results of Lemaire [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), pp. 91–103], Jost [J. Reine Angew. Math. 359 (1985), pp. 37-54], Schoen–Yau [Ann. of Math. (2) 110 (1979), pp. 127–142], and Sacks–Uhlenbeck [Trans. Amer. Math. Soc. 271 (1982), pp. 639–652].

Hessian of Hausdorff dimension on purely imaginary directions

Bridgeman–Taylor (Math. Ann. 341 (2008), 927–943) and McMullen (Invent. Math. 173 (2008), 365–425) showed that the Weil–Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi-Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) on the Hitchin component for PSL d ( R ) ${{\mathsf {PSL}}}_d(\mathbb {R})$ . More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)-hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1–51) which includes small complex deformations of Hitchin representations and of Θ $\Theta$ -positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ → PO ( n , 1 ) → PU ( n , 1 ) ${{\Gamma }}\rightarrow {{\mathsf {PO}}}(n,1)\rightarrow {{\mathsf {PU}}}(n,1)$ is positive definite when Γ ${{\Gamma }}$ is co-compact in PO ( n , 1 ) ${{\mathsf {PO}}}(n,1)$ (unless n = 2 $n=2$ and the deformation is tangent to X ( Γ , PO ( 2 , 1 ) ) $\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ ).

Corrigendum to Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (J. reine angew. Math. 727 (2017), 269–299)

Abstract There is an error in the proof of Theorem 1.3 of the original article. Despite the problem, it is rigorously proved in joint work of the first two authors and Perales that Theorem 1.3 is true, using recent results of Allen and Perales that extend the work of Allen, Perales, and Sormani.

Isomorphisms among quantum Grothendieck rings and propagation of positivity

Abstract Let ( 𝔤 , 𝗀 ) {\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀 {\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} and 𝒞 𝗀 {\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of 𝔤 {\mathfrak{g}} and 𝗀 {\mathsf{g}} , respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced 𝔤 {\mathfrak{g}} . In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q , t q,t -characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced 𝔤 {\mathfrak{g}} , and for any simple finite-dimensional modules in 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for 𝔤 {\mathfrak{g}} of type B n {\mathrm{B}_{n}} . In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all 𝔤 {\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀 {\mathsf{g}} and subalgebras of the quantum Grothendieck ring of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} .

Intrinsic Flat Convergence of Points and Applications to Stability of the Positive Mass Theorem

We prove results on intrinsic flat convergence of points—a concept first explored by Sormani (Commun Anal Geom 26(6):1317–1373, 2018). In particular, we discuss compatibility with Gromov–Hausdorff convergence of points—a concept first described by Gromov (Inst Hautes Études Sci Publ Math 53:53–73, 1981). We apply these results to the problem of stability of the positive mass theorem in mathematical relativity. Specifically, we revisit the article (Huang et al. in J Reine Angew Math 727:269–299, 2017) on intrinsic flat stability for the case of graphical hypersurfaces of Euclidean space: We are able to fill in some details in the proofs of Theorems 1.4 and Lemma 5.1 of Huang et al. (2017) and strengthen some statements. Moreover, in light of an acknowledged error in the proof of Theorem 1.3 of Huang et al. (2017), we provide an alternative proof that extends recent work of Allen and Perales (Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2020. arXiv:2006.13030 ).

Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities

A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer n ⩾ 1 n \geqslant 1 , if the entrywise application of a smooth function f : ( 0 , ∞ ) → R f : (0,\infty ) \to \mathbb {R} preserves the set of n × n n \times n positive semidefinite matrices with positive entries, then f f and its first n − 1 n-1 derivatives are non-negative on ( 0 , ∞ ) (0,\infty ) . In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108]. In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first n n nonzero Taylor coefficients at individual points rather than on all of ( 0 , ∞ ) (0,\infty ) . A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series 1 / ( 1 − u j v k ) 1 / (1 - u_j v_k) ), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.

Extended Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160)

Article Extended Erratum to A parabolic flow toward solutions of the optimal transportation problem on domains with boundary (J. reine angew. Math. 672 (2012), 127–160) was published on December 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 781).

Sharp nonzero lower bounds for the Schur product theorem

By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product M ∘ N M \circ N of two positive semidefinite matrices M , N M,N is again positive. Vybíral [Adv. Math. 368 (2020), p. 9] improved on this by showing the uniform lower bound M ∘ M ¯ ≥ E n / n M \circ \overline {M} \geq E_n / n for all n × n n \times n real or complex correlation matrices M M , where E n E_n is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 15 (1999), pp. 299–316] and to positive definite functions on groups. Vybíral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of M M , or for M ∘ N M \circ N when N ≠ M , M ¯ N \neq M, \overline {M} . A natural third question is to ask for a tighter lower bound that does not vanish as n → ∞ n \to \infty , i.e., over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybíral’s result to lower-bound M ∘ N M \circ N , for arbitrary complex positive semidefinite matrices M , N M,N . Specifically: we provide tight lower bounds, improving on Vybíral’s bounds. Second, our proof is ‘conceptual’ (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy–Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert–Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs–Krieg–Novak–Vybíral [J. Complexity 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in the error bounds in certain tensor product (integration) problems.

Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101)

Article Erratum to The braided Thompson's groups are of type F∞ (J. reine angew. Math. 718 (2016), 59–101) was published on September 1, 2021 in the journal Journal für die reine und angewandte Mathematik (volume 2021, issue 778).

A Generalization of Strassen’s Theorem on Preordered Semirings

Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

Dominant and global dimension of blocks of quantised Schur algebras

Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and S_q(n,r) with n geqslant r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

Harmonic mappings between singular metric spaces

In this paper, we survey the existence, uniqueness and interior regularity of solutions to the Dirichlet problem associated with various energy functionals in the setting of mappings between singular metric spaces. Based on known ideas and techniques, we separate the necessary analytical assumptions to axiomatizing the theory in the singular setting. More precisely, (1) we extend the existence result of Guo and Wenger (Comm Anal Geom 28(1):89–112, 2020) for solutions to the Dirichlet problem of Korevaar–Schoen energy functional to more general energy functionals in purely singular setting. (2) When Y has non-positive curvature in the sense of Alexandrov (NPC), we show that the ideas of Jost (Calc Var Partial Differ Equ 5(1):1–19, 1997) and Lin (Analysis on singular spaces, collection of papers on geometry, analysis and mathematical physics, World Science Publishers, River Edge, pp 114–126, 1997) can be adapted to the purely singular setting to yield local Hölder continuity of solutions of the Dirichlet problem of Korevaar–Schoen and Kuwae–Shioya. (3) We extend the Liouville theorem of Sturm (J Reine Angew Math 456:173–196, 1994) for harmonic functions to harmonic mappings between singular metric spaces. (4) We extend the theorem of Mayer (Comm Anal Geom 6:199–253, 1998) on the existence of the harmonic mapping flow and solve the corresponding initial boundary value problem. Combing these known ideas, with the more or less standard techniques from analysis on metric spaces based on upper gradients, leads to new results when we consider harmonic mappings from \({{\,\mathrm{RCD}\,}}(K,N)\) spaces into NPC spaces. Similar results for the Dirichlet problem associated with the Kuwae–Shioya energy functional and the upper gradient functional are also derived.

Unraveling the mechanical origin of stable solid electrolyte interphase

Summary The capability of a solid electrolyte interphase (SEI) in accommodating deformation is critical to the electrode integrity. However, the nanoscale thickness and environmental sensitivity of SEI make it challenging to characterize multiple mechanical parameters and identify an appropriate predictor for such capability. Here, we develop a feasible atomic force microscopy (AFM)-based nanoindentation test that circumvents the interference of the anode substrate and allows accurate probing of Young's modulus and elastic strain limit of SEI. The "maximum elastic deformation energy" (U) that an SEI can absorb is proposed to predict the stability of the SEI, as successfully demonstrated in Li/K metal anodes. We show that another asset of U is to provide a rapid and effective means to screen proper electrolytes for the stabilization of Sn and Sb microparticle anodes through building high-U SEIs. Overall, this new indicator, U, offers future directions toward rational design of robust SEIs for advanced anodes.

Loewner chains with quasiconformal extensions: an approximation approach

A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation $${{\partial {f_t}\left(z \right)} \over {\partial t}} = \left({z - \tau \left(t \right)} \right)\left({1 - \overline {\tau \left(t \right)} z} \right)p\left({z,\;t} \right){{\partial {f_t}\left(z \right)} \over {\partial z}},$$ where τ: [0, ∞) → \(\overline{\mathbb{D}}\) is measurable and p is called a Herglotz function. In this paper, we will show that if there exists a k ⩽ [0, 1) such that p satisfies $$\left| {p\left({z,\;t} \right) - 1} \right| \le k\left| {p\left({z,\;t} \right) + 1} \right|$$ for all z ⩽ \(\mathbb{D}\) and almost all t ⩽ [0, ∞), then, for all t ⩽ [0, ∞), ft has a k-quasiconformal extension to the whole Riemann sphere. The radial case (τ = 0) and the chordal case (τ = 1) have been proven by Becker [J. Reine Angew. Math., vol. 255 (1972), 23–43] and Gumenyuk and the author [Math. Z., vol. 285 (2017), 1063–1089]. In our theorem, no superfluous assumption is imposed on τ ⩽ \(\overline{\mathbb{D}}\). As a key foundation of the proof is an approximation method using a continuous dependence of evolution families and Loewner chains.

Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals

In David et al. (Adv Math 350:1109–1192, 2019) and David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020), the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli (J Reine Angew Math 325:105–144, 1981) and Alt et al. (Trans Am Math Soc 282:431–461, 1984). In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to Alt and Caffarelli, J Reine Angew Math 325:105–144, 1981; Alt et al., Trans Am Math Soc 282:431–461, 1984; David et al., Adv Math 350:1109–1192, 2019; David and Toro, Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) which deal only with the “Laplacian” setting). We prove Lipschitz regularity up to, and across, the free boundary, fully generalizing the results of David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) to the variable coefficient setting.

Vector bundles on rational homogeneous spaces.

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur J Math 6:430–452, 2020). In particular, if X is a generalized Grassmannian {mathcal {G}} and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. So we improve the main theorem of Muñoz et al. (J Reine Angew Math (Crelles J) 664:141–162, 2012, Theorem 3.1) when X is a generalized Grassmannian. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert–Mülich–Barth theorem on rational homogeneous spaces.

Interior estimates of harmonic heat flow

Motivated by Giaquinta and Hildebrandt’s regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a [Formula: see text]-regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser’s iteration and Schauder estimate.