XXVI. There too, forever wand'ring near,
Is seen swift-footed Eikthyrner;¹
He on Lærad's foliage feeds,
And annually prolific breeds.
Fast in Hrvergelmer's² tide,
Dew-drops down his antlers glide;
Whence, winding thro' the porous earth,
Augmented rivers take their birth.

¹ EIKTHYRNER, the Stag.
² HRVERGELMER, the father of rivers.

The stag, which over the hall of Heriafauthr,
Standeth, and browseth for food the boughs of Laerath,
Is Esikthyrnir; and into Huergelmer
Droppeth from his horns the limpid dew; thence flow

26. Eikthyrnir the hart is called,
that stands o’er Odin’s hall,
and bits from Lærad’s branches;
from his horns fall
drops into Hvergelmir,
whence all waters rise:-

Oakthorn is the name of the hart that stands on the hall of the Father of Hosts and bites at the boughs of Learad : his horns drip into the Boiling-cauldron [Tartarus], whence come all the rivers on earth. . . .

26. Oak-thorn o'er Valholl stands, the hart,
who gnaws the Shelterer's boughs ;
run drops from his horns into Roaring-kettle
whence flow all floods in the world.

26. Eikthyrnir, ³⁸ the hart on the hall that stands,
eateth off Læráth's limbs;
drops from his horns in Hvergelmir³⁹ fall,
thence wend all the waters their way.
 
³⁸ "Oak Antlers" (?).
³⁹ A well at the foot of Yggdrasil.

26. Oak-Thorn the hart in the hall of All-Father
Who bites at Laerað's boughs:
His horns drip into Hvergelmir,
Whence all waters rise.

Eikthrynir the stag stands on Warfather's hall
and bites off Laerad's branches;
drops fall from his horns down to the well
from which the world's rivers run.
 
down to the well= Hvergelmir, the well at the root of Yggdrasil

26. Eikthyrnir is the hart's name, who stands on Father of Hosts' hall
and grazes Lærad's branches;
and from his horns liquid drips into Hvergelmir,
from thence all waters have their flowing:

26. Oak thorn is the stag's name
that stands on the Father of Armies' hall
and browses from Listener's branches,
while drops from his antlers
drip into Cauldron Roar-
all waters take their ways from
there:

26. ‘Oak-antlered is the name of the hart,
standing on Host-father’s hall,
who bites Lærad’s limbs;
from his horns there drips into Hvergelmir
the source from which all rivers run.

The stag Eikþyrnir stands on the roof of Valhall and eats from the branches of the World-Tree, here called Læraðr. Valhall appears to be depicted as a hall similar to the one described in Völsungasaga, ch. 2. Thus the stag, standing on its roof, can eat from the tree.

Svo er sagt að Völsungur konungur lét gera höll eina ágæta og með þeim hætti að ein eik mikil stóð í höllinni og limar trésins með fögrum blómum stóðu út um ræfur hallarinnar en leggurinn stóð niður í höllina og kölluðu þeir það barnstokk.

"It is said that King Volsung had an excellent palace built in this fashion: a huge tree¹ stood with its trunk in the hall and its branches, with fair blossoms, stretched out through the roof. They called the tree Barnstock.^"2  (Jessie Byock tr.)

Eikþyrnir, the name of the stag, is most commonly translated as 'Oak-Thorn', and taken as a reference to its antlers.

1923 Henry Bellows, The Poetic Edda, “Grimnismol: The Ballad of Grimnir”:
 
26. Eikthyrnir ("The Oak-Thorned," i.e., with antlers, "thorns," like an oak): this animal presumably represents the clouds. The first line, like that of stanza 25, is too long in the original. Lærath: cf. stanza 25, note.

Hvergelmir: according to Snorri, this spring, "the Cauldron-Roaring," was in the midst of Niflheim, the world of darkness and the dead, beneath the third root of the ash Yggdrasil. Snorri gives a list of the rivers flowing thence nearly identical with the one in the poem.

Gylfaginning 4:

Fyrr var þat mörgum öldum en jörð var sköpuð er Niflheimr var gerr, ok í honum miðjum liggr bruðr sá, er Hvergelmir heitir, ok þaðan af falla þær ár, er svá heita: Svöl, Gunnþrá, Fjörm, Fimbulþul, Slíðr ok Hríð, Sylgr ok Ylgr, Víð, Leiptr. Gjöll er næst Helgrindum.

"It was many ages before the earth was shaped that Niflheim was made; and midmost within it lies the well that is called Hvergelmir, from which spring the rivers called Svöl, Gunnthrá, Fjörm, Fimbulthul, Slídr and Hríd, Sylgr and Ylgr, Víd, Leiptr; Gjöll is hard by Hel-gates."

Snorri also paraphrases this verse in Gylfaginning 39:

Enn er meira mark at of hjörtinn Eikþyrni, er stendr á Valhöll ok bítr af limum þess trés, en af hornum hans verðr svá mikill dropi, at niðr kemr í Hvergelmi, ok þaðan af falla þær ár, er svá heita: Síð, Víð, Sækin, Ekin, Svöl, Gunnþró, Fjörm, Fimbulþul, Gípul, Göpul, Gömul, Geirvimul. Þessar falla um ása byggðir. Þessar eru enn nefndar: Þyn, Vín, Þöll, Höll, Gráð, Gunnþráin, Nyt, Nöt, Nönn, Hrönn, Vína, Vegsvinn, Þjóðnuma."

"Even more worthy of note is the hart Eikthyrnir, which stands in Valhall and bites from the limbs of the tree; and from his horns distils such abundant exudation that it comes down into Hvergelmir, and from thence fall those rivers called thus: Síd, Víd, Søkin, Eikin, Svöl, Gunnthrá, Fjörm, Fimbulthul, Gípul, Göpul, Gömul, Geirvimul. Those fall about the abodes of the Æsir; these also are recorded: Thyn, Vín, Thöll, Höll, Grád, Gunnthráin, Nyt, Nöt, Nönn, Hrönn, Vína, Vegsvinn, Thjódnuma."

 The name Eikþyrnir is also found in the þula (name-list) of stags.

Ursula Dronke, The Poetic Edda, Volume III Mythological Poems, Part 1, (1997), p. 48-49:

(1) Heimdall as watchman. Beneath the holy tree that is heiðvanr — 'accustomed to bright mead' from the well at its foot— the völva 'knows' that the miraculous hearing (hljóð) of Heimdallr is couched. Tree of the world, he is the divine sentinel. To catch all sounds, his 'ear' is to the ground, where the tree-roots in the nine realms of the underworld attend the vibrations of the giants' coming.

She knows that beneath the tree the sound (hljóð) of Heindallr's horn lies silent. It will be heard at Ragnarök (45). Until then the horn stays a drinking horn for the mead of wisdom, by the well at the foot of the tree. Sigrdrífa speaks of this horn, dripping runes, in a distinct mythological context in Sigrdrífumál 13.

The poet here employs the two senses of hljóð to recall both Heimdallr's 'hearing' and his 'sound'— that is, his echoing horn. Like any good sentinel, he hears and is heard.

Völuspá 27/2:

hljóð: both 'hearing' (cf. Vsp 1/1) and 'sound' (klukku hljóð, 'sound of a bell'). In OE Riddle 4 (ASPR III. 187) the horn is described as a wind instrument, swallowing wind from some man's breast, summoning a troop to war, and also as a drinking vessel, filled by a courtly girl. I think it likely that in his use of hljóð in the sense 'sound' the poet of Vsp is alluding to Heimdallr's horn both as a sounding horn and as a drinking horn, as in the OE riddle (and, indeed, as Snorri understands it: Mimir is full of knowledge 'because he drinks from the well out of the horn Gjallarhorn', SnE 22). So hljóð, 'sound' i.e. 'horn', becomes associated with the well of mead (from which Odinn takes his drink of wisdom, cf. 27/5-7,28/7-12: a myth of which Sigrdr 13/6-10, including a dripping horn, seems to be a variant), as well as with Ragnarök, when the horn's hljóð must sound (45). As to the acute hearing of the world tree, Heimdallr, huntsmen say that birds in a tree will hear a distant gunshot much sooner than a man standing near them (and fly off), because the ground carries the vibrations more swiftly than the air, and they are communicated via the tree's roots to the birds.”

“Some Thoughts on Völuspá” by Paul Schach from Edda, A Collection of Essays (1983), p. 99:

 “Heimdallar hljóð has aroused much speculation. Snorri seems to have confused this word with Gjallarhorn, but there is otherwise no attestation of the use of hljóð in the sense “horn” in Icelandic. Nordal espouses the view set forth by Detter, Heinzel and Höckert that hljóð in this stanza is used in its original meaning of  ‘hearing.’ Olafur Breim glosses hljóð with hlust ‘auditory passage’ and heyrn ‘sense of hearing, ear’ Guðni Jónsson lists only heyrn. Turville-Petre (1964, p. 149) suggests that Heimdallr’s hearing ‘may be conceived in concrete form, as one of Heimdallr’s ears.’